Statistics Chapter 3 Homework

3.5 Which measures are used in the​ five-number summary? Select all that apply. A. Maximum value B. Third quartile C. Variance D. First quartile E. Mean F. Mode G. Standard deviation H. Median I. Interquartile range J. Minimum value

A. Maximum value B. Third quartile D. First quartile H. Median J. Minimum value Note: The​ five-number summary of a data set consists of the smallest data​ value, the first quartile Q1​, the​ median, the third quartile Q3​, and the largest data value.

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 12. Use the empirical rule to determine the following. ​(a) What percentage of people has an IQ score between 88 and 112​? ​(b) What percentage of people has an IQ score less than 76 or greater than 124​? ​(c) What percentage of people has an IQ score greater than 136​?

Recall that if data have a distribution that is bell​ shaped, the empirical rule can be used to determine the percentage of data that lie within k standard deviations of the mean. The empirical rule says that if a distribution is roughly bell​ shaped, the following is true. ​· Approximately​ 68% of the data will lie within 1 standard deviation of the mean. ​· Approximately​ 95% of the data will lie within 2 standard deviations of the mean. ​· Approximately​ 99.7% of the data will lie within 3 standard deviations of the mean. To help organize the empirical rule and make the analysis​ easier, draw a​ bell-shaped curve, as shown to the right. The line in the center of the curve represents the mean. The other lines are each​ 1, 2, and 3 standard deviations away from the mean.. A bell curve is above a horizontal axis labeled from 64 to 136 in intervals of 12. Seven vertical line segments extend from the horizontal axis to the bell curve at regular intervals, dividing the area under the bell curve into eight regions. From left to right, the vertical line segments are labeled mu minus 3 sigma, mu minus 2 sigma, mu minus sigma, mu, mu plus sigma, mu plus 2 sigma, and mu plus 3 sigma. The middle vertical line segment extends from the value 100. Now​ let's label the lines and the areas in the graph. The value of the indicated line is calculated below. μ+σ=112 A bell curve is above a horizontal axis labeled from 64 to 136 in intervals of 12. Seven vertical line segments extend from the horizontal axis to the bell curve at regular intervals, dividing the area under the bell curve into eight regions. From left to right, the vertical line segments are labeled mu minus 3 sigma, mu minus 2 sigma, mu minus sigma, mu, mu plus sigma, mu plus 2 sigma, and mu plus 3 sigma. The middle vertical line segment extends from the value 100. The fifth vertical label from the left is highlighted and extends from a value labeled with a question mark. ​Similarly, the value of the first line of the left​ side, which is 1 standard deviation​ away, is 100−12=88. ​Also, since​ 68% of the data will lie within 1 standard deviation of the​ mean, half of​ this, 34%, will lie on each side of the mean. A bell curve is above a horizontal axis labeled from 64 to 136 in intervals of 12. Seven vertical line segments extend from the horizontal axis to the bell curve at regular intervals, dividing the area under the bell curve into eight regions. From left to right, the vertical line segments are labeled mu minus 3 sigma, mu minus 2 sigma, mu minus sigma, mu, mu plus sigma, mu plus 2 sigma, and mu plus 3 sigma. The third, fourth, and fifth vertical line segments extend from the values 88, 100, and 112, respectively. The middle two regions are labeled 34%. Now label the lines that are 2 standard deviations away from the mean. μ−2σ is 100−2(12)=76. Similarly μ+2σ is 124. The shaded areas are found using the empirical rule. Since​ 95% of the data lies within 2 standard deviations of the mean and​ 68% lies within 1 standard​ deviation, these two areas will add up to 95−68=​27%. The two areas are​ equivalent, so each represents 27/2=13.5% of the data. A bell curve is above a horizontal axis labeled from 64 to 136 in intervals of 12. Seven vertical line segments extend from the horizontal axis to the bell curve at regular intervals, dividing the area under the bell curve into eight regions. From left to right, the vertical line segments extend from an unlabeled value, 76, 88, 100, 112, 124, and an unlabeled value, and are labeled mu minus 3 sigma, mu minus 2 sigma, mu minus sigma, mu, mu plus sigma, mu plus 2 sigma, and mu plus 3 sigma. The third and sixth regions from the left are shaded. The middle two regions are labeled 34%. Now​ let's finish labeling the graph. The lines that are 3 standard deviations away from the mean are μ−3σ=64 and μ+3σ=136. The shaded areas are found using the empirical rule. Since​ 99.7% of the data lies within 3 standard deviations of the mean and​ 95% lie within 2 standard​ deviations, these two areas will add up to 99.7−95=4.7%. The two areas are​ equivalent, so each represents 4.7/2=2.35% of the data. A bell curve is above a horizontal axis labeled from 64 to 136 in intervals of 12. Seven vertical line segments extend from the horizontal axis to the bell curve at regular intervals, dividing the area under the bell curve into eight regions. From left to right, the vertical line segments extend from the values 64, 76, 88, 100, 112, 124, and 136, and are labeled mu minus 3 sigma, mu minus 2 sigma, mu minus sigma, mu, mu plus sigma, mu plus 2 sigma, and mu plus 3 sigma. From left to right, the first region is unlabeled and unshaded, the second region is unlabeled and shaded, the third, fourth, fifth, and sixth regions are labeled 13.5%, 34%, 34%, and 13.5%, respectively, the seven region is unlabeled and shaded, and the eighth region is unlabeled and unshaded. ​Finally, the leftover areas represents 100−99.7=0.3% of the data.​ Thus, each area represents​ 0.15% of the data. A bell curve is above a horizontal axis labeled from 64 to 136 in intervals of 12. Seven vertical line segments extend from the horizontal axis to the bell curve at regular intervals, dividing the area under the bell curve into eight regions. From left to right, the vertical line segments extend from the values 64, 76, 88, 100, 112, 124, and 136, and are labeled mu minus 3 sigma, mu minus 2 sigma, mu minus sigma, mu, mu plus sigma, mu plus 2 sigma, and mu plus 3 sigma. From left to right, the regions are labeled 0.15%, 2.35%, 13.5%, 34%, 34%, 13.5%, 2.35%, and 0.15%. Now use the graph to answer the questions. ​(a) What percentage of people has an IQ score between 88 and 112​? To answer this​ question, add up the correct percentages from the graph. The graph indicates that ​34%+​34% of people have an IQ between 88 and 112​, so the correct answer is 68​%. A bell curve is above a horizontal axis labeled from 64 to 136 in intervals of 12. Seven vertical line segments extend from the horizontal axis to the bell curve at regular intervals, dividing the area under the bell curve into eight regions. From left to right, the vertical line segments extend from the values 64, 76, 88, 100, 112, 124, and 136, and are labeled mu minus 3 sigma, mu minus 2 sigma, mu minus sigma, mu, mu plus sigma, mu plus 2 sigma, and mu plus 3 sigma. From left to right, the regions are labeled 0.15%, 2.35%, 13.5%, 34%, 34%, 13.5%, 2.35%, and 0.15%. ​(b) What percentage of people has an IQ score less than 76 or greater than 124​? The graph indicates that ​0.15%+​2.35% score less than 76 and the same percent score greater than 124. So the correct answer is ​2(0.15+​2.35)=5​%. curve is above a horizontal axis labeled from 64 to 136 in intervals of 12. Seven vertical line segments extend from the horizontal axis to the bell curve at regular intervals, dividing the area under the bell curve into eight regions. From left to right, the vertical line segments extend from the values 64, 76, 88, 100, 112, 124, and 136, and are labeled mu minus 3 sigma, mu minus 2 sigma, mu minus sigma, mu, mu plus sigma, mu plus 2 sigma, and mu plus 3 sigma. From left to right, the regions are labeled 0.15%, 2.35%, 13.5%, 34%, 34%, 13.5%, 2.35%, and 0.15%. ​(c) What percentage of people has an IQ score greater than 136​? From the graph we can see that 0.15​% of the people has an IQ greater than 136. A bell curve is above a horizontal axis labeled from 64 to 136 in intervals of 12. Seven vertical line segments extend from the horizontal axis to the bell curve at regular intervals, dividing the area under the bell curve into eight regions. From left to right, the vertical line segments extend from the values 64, 76, 88, 100, 112, 124, and 136, and are labeled mu minus 3 sigma, mu minus 2 sigma, mu minus sigma, mu, mu plus sigma, mu plus 2 sigma, and mu plus 3 sigma. From left to right, the regions are labeled 0.15%, 2.35%, 13.5%, 34%, 34%, 13.5%, 2.35%, and 0.15%.

A highly selective boarding school will only admit students who place at least 1.5 standard deviations above the mean on a standardized test that has a mean of 100 and a standard deviation of 24. What is the minimum score that an applicant must make on the test to be​ accepted? The minimum score that an applicant must make on the test to be accepted is________

136

3.5 The data to the right represent the number of chocolate chips per cookie in a random sample of a name brand and a store brand. Complete parts ​(a) to ​(c) below. Name Brand 23 25 25 22 27 20 29 33 30 22 21 26 20 Store Brand 15 20 16 20 33 27 28 24 18 26 31 21 23 ​(a) Draw​ side-by-side boxplots for each brand of cookie. Label the boxplots​ "N" for the name brand and​ "S" for the store brand. Choose the correct answer below. A. Two stacked horizontal boxplots are above a horizontal number line from 10 to 40 in increments of 2. A boxplot labeled "N" consists of a box extending from 22 to 28 with a vertical line segment through the box at 25 and two horizontal line segments extending from the left and right sides of the box to 15 and 38, respectively. A boxplot labeled "S" consists of a box extending from 19 to 28 with a vertical line segment through the box at 23 and two horizontal line segments extending from the left and right sides of the box to 13 and 36, respectively. All values are approximate. B. Two stacked horizontal boxplots are above a horizontal number line from 10 to 40 in increments of 2. A boxplot labeled "N" consists of a box extending from 20 to 33 with a vertical line segment through the box at 28 and two horizontal line segments extending from the left and right sides of the box to 19 and 38, respectively. A boxplot labeled "S" consists of a box extending from 15 to 33 with a vertical line segment through the box at 28 and two horizontal line segments extending from the left and right sides of the box to 11 and 39, respectively. All values are approximate. C. Two stacked horizontal boxplots are above a horizontal number line from 10 to 40 in increments of 2. A boxplot labeled "N" consists of a box extending from 22 to 28 with a vertical line segment through the box at 25 and two horizontal line segments extending from the left and right sides of the box to 20 and 33, respectively. A boxplot labeled "S" consists of a box extending from 19 to 28 with a vertical line segment through the box at 23 and two horizontal line segments extending from the left and right sides of the box to 15 and 33, respectively. All values are approximate. ​(b) Does there appear to be a difference in the number of chips per​ cookie? A. Yes. The name brand appears to have more chips per cookie. B. Yes. The store brand appears to have more chips per cookie. C. No. There appears to be no difference in the number of chips per cookie. D. There is insufficient information to draw a conclusion. ​(c) Does one brand have a more consistent number of chips per​ cookie? A. Yes. The store brand has a more consistent number of chips per cookie. B. No. Both brands have roughly the same number of chips per cookie. C. Yes. The name brand has a more consistent number of chips per cookie. D. There is insufficient information to draw a conclusion.

(a)C. Two stacked horizontal boxplots are above a horizontal number line from 10 to 40 in increments of 2. A boxplot labeled "N" consists of a box extending from 22 to 28 with a vertical line segment through the box at 25 and two horizontal line segments extending from the left and right sides of the box to 20 and 33, respectively. A boxplot labeled "S" consists of a box extending from 19 to 28 with a vertical line segment through the box at 23 and two horizontal line segments extending from the left and right sides of the box to 15 and 33, respectively. All values are approximate. Note: Find the​ five-number summary for both sets of data. Determine the lower and upper fences. Draw a number line long enough to include the maximum and minimum values. Insert vertical lines at Q1​, ​M, and Q3. Enclose these vertical lines in a box. Label the lower and upper fences. Draw a line from Q1 to the smallest data value that is larger than the lower fence. Draw a line from Q3 to the largest data value that is smaller than the upper fence. Any data values less than the lower fence or greater than the upper fence are outliers and are marked with an asterisk. (b)A. Yes. The name brand appears to have more chips per cookie. Note: Use the boxplots to compare the medians of the two brands. Use that comparison to determine if one has more chips per cookie. (c)C. Yes. The name brand has a more consistent number of chips per cookie.

3.1 True or​ False: A data set will always have exactly one mode. Choose the correct answer below. a. False b. True

a. False Note: The mode of a variable is the most frequent observation of the variable that occurs in the data set. To compute the​ mode, tally the number of observations that occur for each data value. The data value that occurs most often is the mode. A set of data can have no​ mode, one​ mode, or more than one mode. If no observation occurs more than​ once, the data have no mode.

3.5 In a​ boxplot, if the median is to the left of the center of the box and the right whisker is substantially longer than the left​ whisker, the distribution is skewed ________

right Note: In a​ boxplot, if the median is to the left of the center of the box and the right whisker is substantially longer than the left​ whisker, the distribution is skewed right

3.4 The​ _______ represents the number of standard deviations an observation is from the mean.

z-score Note: The​ z-score represents the distance that a data value is from the mean in terms of the number of standard deviations. It is found by subtracting the mean from the data value and dividing the result by the standard deviation. The​ z-score is unitless. It has mean 0 and standard deviation 1.

3.1 Find the population mean or sample mean as indicated. ​Sample: 20​, 15​, 2​, 13​, 25 Select the correct choice below and fill in the answer box to complete your choice. A. x=___ B. μ=____

A. x= 15 Note: Added all together then divided by 5 since there are 5 numbers

3.2 Match the histograms to the summary statistics given. Mean Median Standard Deviation

Note: Compare the given means and medians to the centers of the data in each histogram. When comparing two​ populations, the larger the standard​ deviation, the more dispersion the distribution​ has, provided that the variable of interest from the two populations has the same unit of measure.

3.2 Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 15. Use the empirical rule to determine the following. ​(a) What percentage of people has an IQ score between 55 and 145​? ​(b) What percentage of people has an IQ score less than 55 or greater than 145​? ​(c) What percentage of people has an IQ score greater than 130​?

​(a)99.7​% ​(Type an integer or a​ decimal.) ​(b) .3​% ​(Type an integer or a​ decimal.). ​(c) 2.5​% ​(Type an integer or a​ decimal.)

3.3 In​ Marissa's calculus​ course, attendance counts for 5​% of the​ grade, quizzes count for 10​% of the​ grade, exams count for 60​% of the​ grade, and the final exam counts for 25​% of the grade. Marissa had a 100​% average for​ attendance, 89​% for​ quizzes, 82​% for​ exams, and 81​% on the final. Determine​ Marissa's course average. ​Marissa's course average is _____%. ​(Type an integer or a decimal. Do not​ round.)

83.35

3.3 Clarissa has just completed her second semester in college. She earned a grade of D in her 4​-hour discrete math course, a grade of A in her 2​-hour economics ​course, a grade of B in her 1​-hour physics course, and a grade of B in her 4​-hour speech writing course. Assuming that A equals 4​points, B equals 3​points, C equals 2​points, D equals 1​point, and F is worth no​points, determine Clarissa​'s grade-point average for the semester.

Clarissa​'s grade point average is 2.45

3.1 A professor has recorded exam grades for 30 students in his​ class, but one of the grades is no longer readable. If the mean score on the exam was 82 and the mean of the 29 readable scores is 84​, what is the value of the unreadable​ score? The value of the unreadable score is ______

24

3.4 _______ divide data sets in fourths.

Quartiles Note: The most common percentiles are quartiles. Quartiles divide data sets into​ fourths, or four equal parts. The first​ quartile, denoted Q1​, divides the bottom​ 25% of the data from the top​ 75%. Therefore, the first quartile is equivalent to the 25th percentile. The second​ quartile, Q2​, divides the bottom​ 50% of the data from the top​ 50%; it is equivalent to the 50th percentile or the median.​ Finally, the third​ quartile, Q3​, divides the bottom​ 75% of the data from the top​ 25%; it is equivalent to the 75th percentile.

3.3 Stan and Francine want to make perfume. In order to get the right balance of ingredients for their tastes they bought 3 ounces of rose oil at $3.03 per ounce, 3 ounces of ginger essence for $3.85 per ounce, and 5 ounces of black currant essence for $3.84 per ounce. Determine the cost per ounce of the perfume. The cost per ounce of the perfume is ​$_______. ​(Round to the nearest​ cent.)

3.62

3.3 ​Recently, a random sample of 25-34 year olds was​ asked, "How much do you currently have in​ savings, not including retirement​ savings?" The data in the table represent the responses to the survey. Approximate the mean and standard deviation amount of savings. Click the icon to view the frequency distribution for the amount of savings.

The sample mean amount of savings is ​$230 ​(Round to the nearest dollar as​ needed.) The sample standard deviation is ​$230 ​(Round to the nearest dollar as​ needed.)

3.2 Compute the range and sample standard deviation for strength of the concrete​ (in psi). 3910​, 4050​, 3200​, 3100​, 2920​, 3810​, 4050​, 4050 1. The range is ______ psi. 2. s= ________ psi

1. The range is 1130 psi. 2. s=479.5 psi​ (Round to one decimal place as​ needed.)

3.2 Select the correct choice that completes the sentence below. The sum of the deviations about the mean _________

always equals zero. Note: the population mean is μ= (∑xi) / N​, the sum of the deviations about the mean ∑(xi−μ) can be rewritten as ∑xi−Nμ​, or ∑xi−∑xi​, which is zero.​ Similarly, since the sample mean is x=(∑xi) / n​, the sum of the deviations about the mean ∑(xi−x) be rewritten as ∑xi−nx​, or ∑xi−∑xi, which is zero.

3.5 The data represent the age of world leaders on their day of inauguration. Find the​ five-number summary, and construct a boxplot for the data. Comment on the shape of the distribution. 52 56 63 64 61 48 58 47 48 50 62 46 55 49 55 (a)The​ five-number summary is (b)Choose the correct description of the shape of the distribution. A. The distribution is skewed to the left. B. The distribution is skewed to the right. C. The distribution is roughly symmetric. D. The shape of the distribution cannot be determined from the boxplot.

(a) 46​, 48​, 55​, 61​, 64. (b) C. The distribution is roughly symmetric.

3.3 The following data represent the​ high-temperature distribution for a summer month in a city for some of the last 130 years. Treat the data as a population. Complete parts​ (a) through​ (c). Temperature ​50-59 ​60-69 ​70-79 ​80-89 ​90-99 ​100-109 Days 5 306 1414 1488 405 9 ​(a) Approximate the mean and standard deviation for temperature. ​(b) Use the frequency histogram of the data to verify that the distribution is bell shaped. ​ a. Yes, the frequency histogram of the data is bell shaped. ​b. No, the frequency histogram of the data is not bell shaped. A histogram has a horizontal axis labeled from 50 to 110 and a vertical axis labeled from 0 to 1700 in increments of 170. Six vertical bars with class width 10 are plotted with approximate heights as follows, with the class listed first and the height listed second: 50 to 60, 10; 60 to 70, 310; 70 to 80, 1410; 80 to 90, 1490; 90 to 100, 410; 100 to 110, 10. ​(c) According to the empirical​ rule, 95% of days in the month will be between what two​ temperatures?

a. μ=80.5 ​(Round to one decimal place as​ needed.) σ=8.1 ​(Round to one decimal place as​ needed.) b. a. Yes, the frequency histogram of the data is bell shaped. c. 64.3 and 96.7 ​(Round to one decimal place as needed. Use ascending​ order.)

3.1 What does it mean if a statistic is​ resistant? Choose the correct answer below. A. Extreme values​ (very large or​ small) relative to the data do not affect its value substantially. B. Extreme values​ (very large or​ small) relative to the data affect its value substantially. C. Changing particular data values affects its value substantially. D. An estimate of its value is extremely close to its actual value.

A. Extreme values​ (very large or​ small) relative to the data do not affect its value substantially. Note: A statistic is resistant if it is not sensitive to extreme values.

3.2 Which of the following measures of dispersion is​ resistant, if​ any? Select all that apply. A. Variance B. Range C. Standard deviation D. None of these measures of dispersion is resistant.

D. None of these measures of dispersion is resistant. Note : A numerical summary of data is said to be resistant if extreme values​(very large or​small) relative to the data do not affect its value substantially. The range is not​resistant, because it is computed using only the largest and smallest values in the data set. The standard deviation and variance are not​resistant, since an extreme value would have a very large deviation from the mean.

3.3 A sample of college students was asked how much they spent monthly on cell phone plans. Approximate the mean for the cost. Monthly cell phone plan cost​ ($) : Number of students ​10.00-19.99 : 9 ​20.00-29.99 : 14 ​30.00-39.99 : 20 ​40.00-49.99 : 16 ​50.00-59.99 : 10 The mean for the cost is _________ ​(Round to the nearest cent as​ needed.)

​$35.58

3.2 Each of the following three data sets represents the IQ scores of a random sample of adults. IQ scores are known to have a mean and median of 100. For each data​ set, determine the sample standard deviation. Then recompute the sample standard deviation assuming that the individual whose IQ is 104 is accidentally recorded as 140. For each sample​ size, state what happens to the standard deviation. Comment on the role that the number of observations plays in resistance. Click the icon to view the three data sets with the misrecorded data value.

For each data​ set, compute the standard deviation. Use technology to find the sample standard​ deviation, s, of the sample of size 5. s≈9.4 Use technology to find the sample standard​ deviation, s, of the sample of size 12. s≈8.6 Use technology to find the sample standard​ deviation, s, of the sample of size 30. s≈9.3 For each data​ set, determine the standard deviation​ again, this time assuming that the individual whose IQ is 104 is accidently recorded as 140. The new table reflects this change. Use technology to find the sample standard​ deviation, s, of the sample of size 5. s≈17.1 Use technology to find the sample standard​ deviation, s, of the sample of size 12. s≈13.5 Use technology to find the sample standard​ deviation, s, of the sample of size 30. s≈11.3 The table below summarizes the standard deviation of the three sample sizes and the new standard deviation with the misrecorded score. Sample size: Standard deviation: New standard deviation 5: 9.4: 17.1 12: 8.6: 13.5 30: 9.3: 11.3 For each sample​ size, the standard deviation increased. The table below summarizes the difference between the standard deviation of the sample sizes with and without the misrecorded score. Sample size: Difference between standard deviations 5: 17.1−9.4=7.7 12: 13.5−8.6=4.9 30: 11.3−9.3=2.0 As the sample size​ increases, the impact of the misrecorded data on the standard deviation decreases.

3.5 (a) Identify the shape of the​ distribution, and​ (b) determine the​ five-number summary. Assume that each number in the​ five-number summary is an integer. A horizontal boxplot is above a number line from 0 to 20 in increments of 1 and consists of a box extending from 2 to 8 with a vertical line segment through the box at 4 and two horizontal line segments extending from the left and right sides of the box to 0 and 19, respectively. a. Choose the correct answer below for the shape of the distribution. A. The distribution is skewed right. B. The distribution is roughly symmetric. C. The distribution is skewed left. D. The shape of the distribution cannot be determined from the boxplot. b. The​ five-number summary is _______, ________, _______, ________, ________

a. A. The distribution is skewed right. b. 0, 2, 4, 8, 19 Note: The​ five-number summary consists of the smallest number in the data​ set, the first​ quartile, the​ median, the third​ quartile, and the largest number in the data set.

3.2 Complete the paragraph below. The standard deviation is used in conjunction with the​ ______ to numerically describe distributions that are bell shaped. The​ ______ measures the center of the​ distribution, while the standard deviation measures the​ ______ of the distribution.

mean, mean, spread Note: Recall that the mean of a variable is computed by determining the sum of all the values of the variable in the data set and dividing by the number of observations. The median of a variable is the value that lies in the middle of the data when arranged in ascending order. The mode of a variable is the most frequent observation of the variable that occurs in the data set.​ Also, the range of a variable is the difference between the largest data value and the smallest data value. The variance is average squared deviation about the​ mean, while the standard deviation describes how​ far, on​ average, each observation is from the mean. Note that the standard deviation and the mean are the most popular methods for numerically describing the distribution of a variable. This is because these two measures are used for most types of statistical inference. Recall that the mean of a variable is computed by determining the sum of all the values of the variable in the data set and dividing by the number of observations. The standard deviation describes how​ far, on​ average, each observation is from the mean. Note that the standard deviation and the mean are the most popular methods for numerically describing the distribution of a variable. This is because these two measures are used for most types of statistical inference.​ Therefore, the standard deviation is used in conjunction with the mean to numerically describe distributions that are bell shaped and symmetric. The mean measures the center of the​ distribution, while the standard deviation measures the spread of the distribution.

3.2 Find the population variance and standard deviation. 6​, 15​, 27​, 33​, 39 1. Choose the correct answer below. Fill in the answer box to complete your choice. ​(Type an integer or a decimal. Do not​ round.) A.s^2=____ B.σ^2=____ 2. Choose the correct answer below. Fill in the answer box to complete your choice. ​(Type an integer or a decimal. Do not​ round.) A. σ=___ B. s=___

1. b.σ^2 = 144 2. a.σ=12 Note: The population variance of a variable is the sum of the squared deviations about the population mean divided by the number of observations in the​ population, N. That​ is, it is the mean of the squared deviations about the population mean. The population variance of a variable is given by the formula

3.1 A histogram of a set of data indicates that the distribution of the data is skewed right. Which measure of central tendency will likely be​ larger, the mean or the​ median? Why? Choose the correct answer below. A. The mean will likely be larger because the extreme values in the left tail tend to pull the mean in the opposite direction of the tail. B. The median will likely be larger because the extreme values in the right tail tend to pull the median in the direction of the tail. C. The mean will likely be larger because the extreme values in the right tail tend to pull the mean in the direction of the tail. D. The median will likely be larger because the extreme values in the left tail tend to pull the median in the opposite direction of the tail.

C. The mean will likely be larger because the extreme values in the right tail tend to pull the mean in the direction of the tail. Note: When data are either skewed left or skewed​ right, there are extreme values in the​ tail, which tend to pull the mean in the direction of the tail. If the distribution of the data is skewed​ right, there are large observations in the right tail. These observations tend to increase the value of the​ mean, while having little effect on the median.

3.3 Rose has just completed her second semester in college. She earned a grade of B in her​ 2-hour differential equations​ course, a grade of A in her​ 1-hour history​ course, a grade of C in her​ 3-hour quantum physics​ course, and a grade of A in her​ 5-hour quantum laboratory course. Assuming that A equals 4​ points, B equals 3​ points, C equals 2​ points, D equals 1​ point, and F is worth no​ points, determine​ Rose's grade-point average for the semester.

Finding​ Rose's grade point average is just like finding the weighted mean of her grades. Each course grade is a category and the number of course hours for that grade is the frequency. Remember that the weighted mean is given by the formula where xi is the midpoint or value of the ith​ class, wi is the weight on the ith​ class, and n is the number of classes. The third row of the table is filled in with the numerical course grade values. The fourth row shows the course hours. ​Now, total the number of course hours in the fourth row. Σwi = 2+1+3+5 = 11 For the weighted mean​ formula, multiply each xi by the corresponding wi. Σxiwi = ​(3•​2)+​(4•​1)+​(2•​3)+​(4•​5) = 6+4+6+20 Find the sum of the xiwi. Σxiwi = 36 Now use the formula given earlier to compute the weighted mean. xw= (Σxiwi) / Σwi =36 / 11 = 3.27 ​Therefore, Rose's grade point average is 3.27.

3.4 The accompanying data represent the pulse rates​ (beats per​ minute) of nine students. Treat the nine students as a population. Compute the​ z-scores for all the students. Compute the mean and standard deviation of these​ z-scores. Click the icon to view the data table. (a) Compute the​ z-scores for all the students. Complete the table. ​(Round to the nearest hundredth as​ needed.) (b) Compute the mean of these​ z-scores. The mean of the​ z-scores is_____. ​(Round to the nearest tenth as​ needed.) (c) Compute the standard deviation of these​ z-scores. The standard deviation of the​ z-scores is _____. ​(Round to the nearest tenth as​ needed.)

(a) Student: z-score Student 1 0.47 Student 6 1.01 Student 2 −1.53 Student 7 1.01 Student 3 −1.66 Student 8 −0.59 Student 4 1.14 Student 9 0.07 Student 5 0.07 (b) 0.0 (c) 1.0

3.2 Example The accompanying data represent the weights​ (in grams) of a random sample of 48​ M&M plain candies. Complete parts​ (a) through​ (f). Click the icon to view the weights of the​ M&M plain candies.

​(a) Determine the sample standard deviation weight. The sample standard​ deviation, s, is obtained by taking the square root of the sample variance. To find the sample standard​ deviation, use technology or either of the formulas shown​ below, where x is the sample mean and x1, x2,...., xn are the n observations in the sample. s=∑x2i−∑x i2nn−1 or s=∑xi−x2n−1 For this​ exercise, use technology to find the sample standard​ deviation, rounding to three decimal places. s=0.036 gram ​(b) On the basis of the accompanying​ histogram, comment on the appropriateness of using the Empirical Rule to make any general statements about the weights of​ M&Ms. The Empirical Rule says that if a distribution is roughly bell​ shaped, then approximately​ 68% of the data will lie within 1 standard deviation of the mean. Approximately​ 95% of the data will lie within 2 standard deviations of the mean. Approximately​ 99.7% of the data will lie within 3 standard deviations of the mean. A histogram titled Weight of Plain M and Ms has a horizontal axis labeled Weight in grams from 0.7 to 1 and a vertical axis labeled Frequency from 0 to 30 in increments of 5. Six vertical bars with class width 0.05 are plotted with heights as follows, with the class listed first and the height listed second: from 0.7 to 0.75, 0; from 0.75 to 0.8, 1; from 0.8 to 0.85, 11; from 0.85 to 0.9, 21; from 0.9 to 0.95, 14; from 0.95 to 1, 1. Use this information to comment on the appropriateness of using the Empirical Rule to make any general statements about the weights of the​ M&Ms. ​(c) Use the Empirical Rule to determine the percentage of​ M&Ms with weights between 0.803 and 0.947 gram.​ Hint: x=0.875. To help organize the Empirical Rule and make the analysis​ easier, draw a​ bell-shaped curve, as shown to the right. The line in the center of the curve represents the mean. The other lines are each​ 1, 2, and 3 standard deviations away from the mean. A symmetric bell-shaped curve is plotted over a horizontal axis with a coordinate labeled "0.875." The center and peak of the curve are located at the coordinate 0.875. Seven vertical line segments extend from the axis to the curve and are spaced equally horizontally. From left to right, the first three vertical line segments are labeled "x overbar minus three s," "x overbar minus two s," and "x overbar minus s." The fourth vertical line segment is at the horizontal coordinate 0.875 and is labeled "x overbar." From left to right, the last three vertical line segments are labeled "x overbar plus s," "x overbar plus two s," and "x overbar plus three s." Now label the lines and the areas in the graph. Recall from part​ (a) that s=0.036. Calculate the value of the indicated line 1 standard deviation above the mean. x+s=0.911 A symmetric bell-shaped curve is plotted over a horizontal axis with a coordinate labeled "0.875." The center and peak of the curve are located at the coordinate 0.875. Seven vertical line segments extend from the axis to the curve and are spaced equally horizontally. From left to right, the first three vertical line segments are labeled "x overbar minus three s," "x overbar minus two s," and "x overbar minus s." The fourth vertical line segment is at the horizontal coordinate 0.875 and is labeled "x overbar." From left to right, the last three vertical line segments are labeled "x overbar plus s," "x overbar plus two s," and "x overbar plus three s." The vertical line segment labeled "x overbar plus s" is highlighted in a different color and has a question mark along the horizontal axis at the same horizontal coordinate. ​ Similarly, the value of the first line of the left​ side, which is 1 standard deviation​ away, is 0.875−0.036=0.839. ​Also, since​ 68% of the data will lie within 1 standard deviation of the​ mean, half of​ this, 34%, will lie on each side of the mean. A symmetric bell-shaped curve is plotted over a horizontal axis with three coordinates labeled "0.839," "0.875," and "0.911." The center and peak of the curve are located at the coordinate 0.875. Seven vertical line segments extend from the axis to the curve and are spaced equally horizontally. From left to right, the first two vertical line segments are labeled "x overbar minus three s" and "x overbar minus two s." The third vertical line segment is at the horizontal coordinate 0.839 and is labeled "x overbar minus s." The fourth vertical line segment is at the horizontal coordinate 0.875 and is labeled "x overbar s." The fifth vertical line segment is at the horizontal coordinate 0.911 and is labeled "x overbar plus s." From left to right, the last two vertical line segments are labeled "x overbar plus two s" and "x overbar plus three s." The area under the curve and between the vertical line segments at horizontal coordinates 0.839 and 0.875 is labeled "34 percent." The area under the curve and between the vertical lines at horizontal coordinates 0.875 and 0.911 is also labeled "34 percent." Now label the lines that are 2 standard deviations away from the mean. The point x−2s is 0.875−2(0.036)=0.803. Similarly x+2s is 0.947. Each of the shaded areas represents​ 13.5% of the data. A symmetric bell-shaped curve is plotted over a horizontal axis with five coordinates labeled "0.803," "0.839," "0.875," "0.911," and "0.947." The center and peak of the curve are located at the coordinate 0.875. Seven vertical line segments extend from the axis to the curve and are spaced equally horizontally. From left to right, the first vertical line segment is labeled "x overbar minus three s." From left to right, the next five vertical line segments have the following labels and horizontal coordinates, where the label is listed first and the coordinate is listed second: "x overbar minus two s," 0.803; "x overbar minus s," 0.839; "x overbar," 0.875; "x overbar plus s," 0.911; "x overbar plus two s," 0.947. The rightmost vertical line segment is labeled "x overbar plus three s." The area under the curve and between the vertical line segments at horizontal coordinates 0.803 and 0.839 is shaded. The areas under the curve between the vertical line segments at horizontal coordinates 0.839 and 0.875 and between the vertical line segments at horizontal coordinates 0.875 and 0.911 are labeled "34 percent." The area under the curve and between the vertical line segments at horizontal coordinates 0.911 and 0.947 is shaded. Now​ let's finish labeling the graph. The lines that are 3 standard deviations away from the mean are μ−3σ=0.767 and μ+3σ=0.983. The shaded areas are found using the empirical rule. Since​ 99.7% of the data lies within 3 standard deviations of the mean and​ 95% lie within 2 standard​ deviations, these two areas will add up to 99.7%−95%=4.7%. The two areas are​ equivalent, so each represents 4.7%/2=2.35% of the data. A symmetric bell-shaped curve is plotted over a horizontal axis with five coordinates labeled "0.803," "0.839," "0.875," "0.911," and "0.947." The center and peak of the curve are located at the coordinate 0.875. Seven vertical line segments extend from the axis to the curve and are spaced equally horizontally. From left to right, the first vertical line segment is labeled "x overbar minus three s." From left to right, the next five vertical line segments have the following labels and horizontal coordinates, where the label is listed first and the coordinate is listed second: "x overbar minus two s," 0.803; "x overbar minus s," 0.839; "x overbar," 0.875; "x overbar plus s," 0.911; "x overbar plus two s," 0.947. The rightmost vertical line segment is labeled "x overbar plus three s." The area under the curve and between the vertical line segment labeled "x overbar minus three s" and the vertical line segment at horizontal coordinate 0.803 is shaded. The area under the curve and between the vertical line segments at horizontal coordinates 0.803 and 0.839 is labeled "13.5 percent." The area under the curve between the vertical line segments at horizontal coordinates 0.839 and 0.875 and between the vertical line segments at horizontal coordinates 0.875 and 0.911 are labeled "34 percent." The area under the curve and between the vertical line segments at horizontal coordinates 0.911 and 0.947 is labeled "13.5 percent." The area under the curve and between the vertical line segment at horizontal coordinate 0.947 and the vertical line segment labeled "x overbar minus three s" is shaded. ​Finally, the leftover areas represent 100%−99.7%=0.3% of the data.​ Thus, each area represents​ 0.15% of the data. A symmetric bell-shaped curve is plotted over a horizontal axis with seven coordinates labeled "0.767," "0.803," "0.839," "0.875," "0.911," "0.947," and "0.983." The center and peak of the curve are located at the coordinate 0.875. Vertical line segments extend from the axis to the curve and are spaced equally horizontally at each labeled coordinate. From left to right, the labels for each of these line segments are as follows: "x overbar minus three s," "x overbar minus two s," "x overbar minus s," "x overbar," "x overbar plus s," "x overbar plus two s," and "x overbar plus three s." The area under the curve and to the left of the horizontal coordinate 0.767 is labeled ".15 percent." The area under the curve and between the horizontal coordinates 0.767 and 0.803 is labeled "2.35 percent." The area under the curve and between the horizontal coordinates 0.803 and 0.839 is labeled "13.5 percent." The area under the curve between the horizontal coordinates 0.839 and 0.875 and between the horizontal coordinates 0.875 and 0.911 are labeled "34 percent." The area under the curve and between the horizontal coordinates 0.911 and 0.947 is labeled "13.5 percent." The area under the curve and between the horizontal coordinates 0.947 and 0.983 is labeled "2.35 percent." The area under the curve and to the right of the horizontal coordinate 0.983 is labeled ".15 percent." Use the Empirical Rule to determine the percentage of​ M&Ms with weights between 0.803 and 0.947 gram. The percentage of​ M&Ms is​ 95%. A symmetric bell-shaped curve is plotted over a horizontal axis with seven coordinates labeled "0.767," "0.803," "0.839," "0.875," "0.911," "0.947," and "0.983." The center and peak of the curve are located at the coordinate 0.875. Vertical line segments extend from the axis to the curve and are spaced equally horizontally at each labeled coordinate. From left to right, the labels for each of these line segments are as follows: "x overbar minus three s," "x overbar minus two s," "x overbar minus s," "x overbar," "x overbar plus s," "x overbar plus two s," and "x overbar plus three s." The area under the curve and to the left of the horizontal coordinate 0.767 is labeled ".15 percent." The area under the curve and between the horizontal coordinates 0.767 and 0.803 is labeled "2.35 percent." The area under the curve and between the horizontal coordinates 0.803 and 0.839 is labeled "13.5 percent." The area under the curve between the horizontal coordinates 0.839 and 0.875 and between the horizontal coordinates 0.875 and 0.911 are labeled "34 percent." The area under the curve and between the horizontal coordinates 0.911 and 0.947 is labeled "13.5 percent." The area under the curve and between the horizontal coordinates 0.947 and 0.983 is labeled "2.35 percent." The area under the curve and to the right of the horizontal coordinate 0.983 is labeled ".15 percent." ​ (d) Determine the actual percentage of​ M&Ms that weigh between 0.803 and 0.947 ​gram, inclusive. To calculate the​ percentage, first count all the​ M&Ms that weigh between 0.803 and 0.947 ​gram, inclusive. There are 46 ​M&Ms that weigh between 0.803 and 0.947. ​ Finally, divide this number by​ 48, and multiply by​ 100, rounding to one decimal place. 46/48×100%=95.8% Use the completed diagram to the right to answer this question. ​(e) Use the Empirical Rule to determine the percentage of​ M&Ms with weights more than 0.911 gram. Find 0.911 on the horizontal axis and then sum the percentages to the right of this value. ​13.5%+​2.35%+​0.15%=​16% A symmetric bell-shaped curve is plotted over a horizontal axis with seven coordinates labeled "0.767," "0.803," "0.839," "0.875," "0.911," "0.947," and "0.983." The center and peak of the curve are located at the coordinate 0.875. Vertical line segments extend from the axis to the curve and are spaced equally horizontally at each labeled coordinate. From left to right, the labels for each of these line segments are as follows: "x overbar minus three s," "x overbar minus two s," "x overbar minus s," "x overbar," "x overbar plus s," "x overbar plus two s," and "x overbar plus three s." The area under the curve and to the left of the horizontal coordinate 0.767 is labeled ".15 percent." The area under the curve and between the horizontal coordinates 0.767 and 0.803 is labeled "2.35 percent." The area under the curve and between the horizontal coordinates 0.803 and 0.839 is labeled "13.5 percent." The area under the curve between the horizontal coordinates 0.839 and 0.875 and between the horizontal coordinates 0.875 and 0.911 are labeled "34 percent." The area under the curve and between the horizontal coordinates 0.911 and 0.947 is labeled "13.5 percent." The area under the curve and between the horizontal coordinates 0.947 and 0.983 is labeled "2.35 percent." The area under the curve and to the right of the horizontal coordinate 0.983 is labeled ".15 percent." ​ (f) Determine the actual percentage of​ M&Ms that weigh more than 0.911 gram. To calculate the​ percentage, first count all the​ M&Ms that weigh more than 0.911 gram. There are 6 ​M&Ms that weigh more than 0.911 gram. ​Finally, divide this number by​ 48, and multiply by​ 100, rounding to one decimal place. 6/48×100%=12.5%

3.2 The accompanying data represent the weights​ (in grams) of a random sample of 48​ M&M plain candies. Complete parts​ (a) through​ (f). Click the icon to view the weights of the​ M&M plain candies. (a) Determine the sample standard deviation weight. ___ gram(s) ​(Round to three decimal places as​ needed.) ​(b) On the basis of the accompanying​ histogram, comment on the appropriateness of using the Empirical Rule to make any general statements about the weights of​ M&Ms. A. The histogram is not approximately​ bell-shaped so the Empirical Rule cannot be used. B. The histogram is approximately​ bell-shaped so the Empirical Rule can be used. C. The histogram is approximately​ bell-shaped so the Empirical Rule cannot be used. D. The histogram is not approximately​ bell-shaped so the Empirical Rule can be used. A histogram titled Weight of Plain M and Ms has a horizontal axis labeled Weight in grams from 0.7 to 1 and a vertical axis labeled Frequency from 0 to 30 in increments of 5. Six vertical bars with class width 0.05 are plotted with heights as follows, with the class listed first and the height listed second: from 0.7 to 0.75, 0; from 0.75 to 0.8, 1; from 0.8 to 0.85, 10; from 0.85 to 0.9, 22; from 0.9 to 0.95, 14; from 0.95 to 1, 1. ​(c) Use the Empirical Rule to determine the percentage of​ M&Ms with weights between 0.803 and 0.947 gram.​ Hint: x=0.875. ______ ​(Type an integer or decimal. Do not​ round.). ​(d) Determine the actual percentage of​ M&Ms that weigh between 0.803 and 0.947 ​gram, inclusive. ______ ​(Round to one decimal place as​ needed.). ​(e) Use the Empirical Rule to determine the percentage of​ M&Ms with weights more than 0.911 gram. _______ ​(Type an integer or decimal. Do not​ round.) ​ (f) Determine the actual percentage of​ M&Ms that weigh more than 0.911 gram. _____ ​(Round to one decimal place as​ needed.)

(a) 0.036 (b) B. The histogram is approximately​ bell-shaped so the Empirical Rule can be used. (c) 95​% ​(Type an integer or decimal. Do not​ round.). ​(d) 95.8​% ​(Round to one decimal place as​ needed.). ​(e)16​% ​(Type an integer or decimal. Do not​ round.) ​ (f) 12.5​%

3.1 Example Find the population mean or sample mean as indicated. ​Sample: 14​, 10​, 11​, 4​, 21

The arithmetic mean of a variable is computed by determining the sum of all the values of the variable in the data​ set, divided by the number of observations. The population arithmetic​ mean, μ​, is computed using all the individuals in a population. The sample arithmetic​ mean, x​, is computed using sample data. For this​ problem, you are finding x because the data set is a sample. To find the sample​ mean, x​, start by calculating the sum of the sample. 14+10+11+4+21=60 Now determine the size of the​ sample, n. There are five values in the data set.​ Therefore, n=5. ​ Finally, to calculate the mean of the sample divide the sum by the size of the sample. x = 60 / 5 = 12

3.4 A highly selective boarding school will only admit students who place at least 3.5 standard deviations above the mean on a standardized test that has a mean of 150 and a standard deviation of 18. What is the minimum score that an applicant must make on the test to be​ accepted?

The​ z-score represents the distance that a data value is from the mean in terms of the number of standard deviations. It is obtained by subtracting the mean from the data value and dividing this result by the standard deviation. The population​ z-score is given by the formula z=x−μ/σ where x is a data​ value, μ is the population​ mean, and σ is the population standard deviation. Substitute the given values of the population mean and the population standard deviation into the formula for the​ z-score. z=x−150 / 18 Since the school will only admit students who place at least 3.5 standard deviations above the mean on the standardized​ test, the​ z-score corresponding to the test score x must be at least 3.5. z=x−150/ 18≥3.5 For the minimum score that an applicant must make on the test to be​ accepted, obtain the equation shown below. . x−150 / 18=3.5 To find the minimum score that an applicant must make on the test to be​ accepted, solve for x. x−150/18=3.5 x=​(3.5)(18)+150 x=213 The minimum score that an applicant must make on the test to be accepted is 213.

3.2 Example Find the sample variance and standard deviation as indicated. 22​, 11​, 6​, 7​, 10

To find the sample​ variance, use technology or either formula shown​ below, where x1, x2, ..., xn are the n observations in the sample and x is the sample mean. s^2= (∑xi^2 − ( [(∑xi)^2] / n)) / n−1 or s^2 = (∑[xi−x]^2) / n−1 For this​ exercise, use the computational formula s^2= (∑xi^2 − ( [(∑xi)^2] / n)) / n−1. To use the computational​ formula, create a table with two columns. Enter the sample data in the first column. In the second​ column, find the square of each data value. Sample​ data, xi : Sample data values​ squared, x2i 22: 22^2=484 11: 11^2=121 6: 36 7: 49 10: 100 Now add the entries in each column to find ∑xi and ∑xi^2. Sample​ data, xi: Sample data values​ squared, x2i 22: 22^2=484 11: 11^2=121 6: 36 7: 49 10: 100 ∑xi=56 : ∑x2i=790 Substitute ∑xi​, ∑x2i​, and n into the computational formula to find the sample​ variance, s2. s^2= (∑xi^2 − ( [(∑xi)^2] / n)) / n−1 = (790−[(56)^2 / 5]) / 5−1 = 40.7 ​Finally, find the standard​ deviation, rounding to one decimal place. Remember that the sample standard​ deviation, s, is obtained by taking the square root of the sample variance. That​ is, s= sqrt s^2. s = sqrt 40.7 =6.4

3.5 example ​(a) Identify the shape of the​ distribution, and​ (b) determine the​ five-number summary. Assume that each number in the​ five-number summary is an integer. A horizontal boxplot is above a number line from 0 to 20 in increments of 1 and consists of a box extending from 13 to 18 with a vertical line segment through the box at 17 and two horizontal line segments extending from the left and right sides of the box to 0 and 20, respectively.

a. Use the horizontal lines on each side of the boxplot and the position of the median to determine whether the shape of the distribution is skewed​ left, right, or is roughly symmetric. If the median is near the center of the box and each horizontal line is of approximately equal​ length, the distribution is roughly symmetric. If the median is to the left of the center of the box or the right line is substantially longer than the left​ line, the distribution is skewed right. If the median is to the right of the center of the box or the left line is substantially longer than the right​ line, the distribution is skewed left. Notice that in the given boxplot the horizontal line on the left is substantially longer than the horizontal line on the right.​ Also, the median is to the right of the center of the box.​ Thus, the shape of the distribution is skewed left. b. The​ five-number summary consists of the smallest and largest numbers in the data​ set, the first​ quartile, the​ median, and the third quartile. Determine each of these values. The smallest number in the data set is the furthest left point on the left horizontal line. Since the left horizontal line extends to​ 0, the smallest number in the data set is 0. Now find the first quartile. The first quartile is where the first black vertical line is located. This makes up the left edge of the boxplot. The first quartile is 13. Next determine the median. The median is located at the black vertical line that lies inside the box. The median is 17. The third quartile is located at the third black vertical line. This is also the right edge of the box. The third quartile is 18. Finally we determine the largest number in the data set. The largest number in the data set is the rightmost point on the right horizontal line. The largest number is 20. ​ Thus, the​ five-number summary is​ 0, 13​, 17​, 18​, and 20.

3.1 Example The following data represent the pulse rates​ (beats per​ minute) of nine students enrolled in a statistics course. Treat the nine students as a population. Complete parts ​(a) through ​(c). Student:Pulse Perpectual Bempah:62 Megan Brooks: 92 Jeff Honeycutt: 80 Clarice Jefferson:80 Crystal Kurtenbach:75 Janette Lantka:80 Kevin McCarthy:78 Tammy Ohm:89 Kathy Wojdya:73

​(a) Determine the population mean pulse. If x1, x2, ..., xN are N observations of a variable from a​ population, then the population​ mean, μ​, is given by the following formula. μ= (x1+x2+...+xN) / N = (∑ xi) / N Though technology can be used to find the​ mean, for the purpose of this​ exercise, use the formula. Use the formula to find μ. Begin by adding all of the values in the population. 62+92+80+80+75+80+78+89+73=709 ​Next, determine the number of observations in this data set. N=9 ​Finally, find the population mean by dividing the sum by the number of observations and rounding to one decimal place. μ = 709 / 9 = 78.8 ​Thus, the population mean pulse is approximately 78.8 beats per minute. ​(b) Determine the sample mean pulse of the following two simple random samples of size 3. Sample​ 1: {Jeff, Janette, Kathy} Sample​ 2: {Clarice, Kevin, Megan} If x1, x2, ..., xn are n observations of a variable from a​ sample, then the sample​ mean, x​, is given by the following formula. x = (x1+x2+...+xn) / n= (∑ xi) / n Either this formula or technology can be used to find the sample mean. Here we will use the formula. The observations in sample 1 are 80​, 80​, and 73. Use the formula to find the sample mean. Begin by adding the values in the sample. 80+80+73=233 ​Finally, find the sample mean by dividing the sum by the number of observations and rounding to one decimal place. x = 233 / 3 = 77.7 ​Thus, the mean pulse of sample 1 is approximately 77.7 beats per minute. The observations in sample 2 are 80​, 78​, and 92. Use the formula to find the sample mean. Begin by adding the values in the sample. 80+78+92=250 Finally, find the sample mean by dividing the sum by the number of observations and rounding to one decimal place. x = 250 / 3 = 83.3 ​Thus, the mean pulse of sample 2 is approximately 83.3 beats per minute. ​(c) Determine if the means of samples 1 and 2​ overestimate, underestimate, or are equal to the population mean. A sample mean overestimates the population mean if it is greater than the population mean. A sample mean underestimates the population mean if it is less than the population mean. The mean pulse rate of sample 1 underestimates the population mean pulse rate because the sample​ mean, 77.7​, is less than the population​ mean, 78.8. The mean pulse rate of sample 2 overestimates the population mean pulse rate because the sample​ mean, 83.3​, is greater than the population​ mean, 78.8.​

3.4 A cellular phone company monitors monthly phone usage. The following data represent the monthly phone use in minutes of one particular customer for the past 20 months. Use the given data to answer parts​ (a) and​ (b). 323 338 542 457 388 422 415 536 454 359 460 407 381 481 437 364 327 395 425 493 ​(a) Determine the standard deviation and interquartile range of the data. ​(b) Suppose the month in which the customer used 323 minutes was not actually that​ customer's phone. That particular month the customer did not use their phone at​ all, so 0 minutes were used. How does changing the observation from 323 to 0 affect the standard deviation and interquartile​ range? What property does this​ illustrate? The standard deviation ______ and the interquartile range _______. (c)What property does this​ illustrate? Choose the correct answer below. a. Resistance b. Dispersion c. Weighted Mean d. Empirical Rule

(a) s=63.59 ​(Round to two decimal places as​ needed.) IQR=86 ​(Type an integer or a​ decimal.) (b)The standard deviation increases and the interquartile range is not affected. (c) a. Resistance Note: A numerical summary of data is said to be resistant if extreme values​ (very large or​ small) relative to the data do not affect its value substantially. Notice that for this data set the standard deviation is not resistant to extreme​ observations, but the interquartile range is resistant.

3.4 The table shows the weekly income of 20 randomly selected​ full-time students. If the student did not​ work, a zero was entered. ​(a) Check the data set for outliers.​ (a) List all the outliers in the given data set. Select the correct choice below and fill in any answer boxes in your choice. A. The​ outlier(s) is/are _______ ​(Use a comma to separate answers as​ needed.) B. There are no outliers. ​(b) Draw a histogram of the data. ​(c) Provide an explanation for any outliers. ​(c) Choose the possible​ reason(s) for any​ outlier(s) below. Select all that apply. A. A student with unusually high income B. Data entry error C. A student providing false information D. None of the above E. There are no outliers.

(a) A. The​ outlier(s) is/are 3174 ​(Use a comma to separate answers as​ needed.) (b) b (c) A,B,C A. A student with unusually high income B. Data entry error C. A student providing false information Note: There are many reasons for an outlier to occur. Some reasons are data entry​ errors, unusual but true​ observations, and data collecting errors.

3.1 Example The following data represent the miles per gallon for a particular make and model car for six randomly selected vehicles. Compute the​ mean, median, and mode miles per gallon. 41.1​, 38.4​, 21.9​, 32.5​, 28.6​, 30.7

The arithmetic mean of a variable is computed by adding all the values of the variable in the data set and dividing by the number of observations. The median of a variable is the value that lies in the middle of the data when arranged in ascending order. The mode of a variable is the most frequent observation of the variable that occurs in the data set. The formula for x​, the arithmetic​ mean, is shown​ below, where x1​, x2​, ​..., xn are n observations of a variable from a sample. x= (x1+x2+•••+xn) / n = (∑ x i) / n While either technology or the formula can be used to find the​ mean, in this​ problem, use technology. Compute the mean. x=32.2 ​Thus, the mean mileage per gallon for the six cars is 32.2. To find the median of a data​ set, first arrange the data in ascending​ order, then determine the number of​ observations, n,​ and, finally, determine the observation in the middle of the data set. If the number of observations is​ odd, then the median is the data value that is exactly in the middle of the data set. That​ is, the median is the observation that lies in the (n+1)/2 position. If the number of observations is​ even, then the median is the mean of the two middle observations in the data set. That​ is, the median is the mean of the observations that lie in the n/2 position and the (n/2)+1 position. While either technology or the process described above can be used to find the​ median, in this​ problem, use technology. Determine the median. M=31.6 ​Thus, the median mileage per gallon for the 6 cars is 31.6. To determine the​ mode, start by determining the frequency of each mileage per gallon. If no observation occurs more than​ once, the data are said to have no mode. Note that no value has a frequency other than 1. Since none of the values are​ repeated, the data set does not have a mode.

3.5 example The data represent the age of world leaders on their day of inauguration. Find the​ five-number summary, and construct a boxplot for the data. Comment on the shape of the distribution. 65 55 58 52 48 62 41 44 50 56 61 55 69 48 44

The​ five-number summary consists of the smallest and largest numbers in the data​ set, the first​ quartile, the​ median, and the third quartile. Be sure to first list the data in ascending order. The data in ascending order are shown below. 41​, 44​, 44​, 48​, 48​, 50​, 52​, 55​, 55​, 56​, 58​, 61​, 62​, 65​, 69 From the list we see that the smallest number in the data set is 41 and the largest number in the data set is 69. Next find the first​ quartile, Q1. Remember that the first quartile is the 25th percentile. Q1=48 Now find the​ median, M. Remember that the median is the 50th percentile. The ordered list is repeated below for reference. 41​, 44​, 44​, 48​, 48​, 50​, 52​, 55​, 55​, 56​, 58​, 61​, 62​, 65​, 69 M=55 Find the third​ quartile, Q3. Remember that the third quartile is the 75th percentile. Q3=61 ​Thus, the​ five-number summary is 41​, 48​, 55​, 61​, and 69. Use the​ five-number summary to construct the boxplot. The first step in drawing a boxplot is to determine the lower and upper fences. The lower fence is given by the​ formula, Lower fence=Q1−​1.5(IQR), and the upper fence is given by the​ formula, Upper fence=Q3+​1.5(IQR). Remember that IQR=Q3−Q1. Compute the lower fence. Lower fence = Q1−​1.5(IQR) = 48−​1.5(61−48​) = 28.5 Compute the upper fence. Upper fence = Q3+​1.5(IQR) = 61+​1.5(61−48​) = 80.5 Now draw vertical lines at Q1​, ​M, and Q3 and enclose them in a box. Also label the lower and upper fence. Temporarily mark the location of the lower and upper fence with brackets​ ([ and​ ]). A horizontal boxplot is above a number line from less than 30 to 80 plus in increments of 2 and consists of a box extending from 48 to 61 with a vertical line segment through the box at 55. A left bracket appears at approximately 29 and a right bracket appears at approximately 80. Next find the smallest data value that is larger than the lower​ fence, 28.5​, and find the largest data value that is smaller than the upper​ fence, 80.5. The ordered list is repeated below for reference. 41​, 44​, 44​, 48​, 48​, 50​, 52​, 55​, 55​, 56​, 58​, 61​, 62​, 65​, 69 The smallest data value that is larger than the lower fence is 41​, and the largest data value that is smaller than the upper fence is 69. Draw a line from Q1 to the smallest data value that is larger than the lower​ fence, and draw a line from Q3 to the largest data value that is smaller than the upper fence. Remove the brackets and adjust the scale to fit the window better. A horizontal boxplot is above a number line from 40 to 70 in increments of 1 and consists of a box extending from 48 to 61 with a vertical line segment through the box at 55 and two horizontal line segments extending from the left and right sides of the box to 41 and 69, respectively. The last step in drawing the boxplot is to mark any outliers with an asterisk. Outliers are data values less than the lower fence or greater than the upper fence. Since there are no outliers in this data​ set, the boxplot is complete. Use the horizontal lines on each side of the boxplot and the position of the median to determine whether the shape of the distribution is skewed​ left, right, or is roughly symmetric. If the median is near the center of the box and each horizontal line is of approximately equal​ length, the distribution is roughly symmetric. If the median is to the left of the center of the box or the right line is substantially longer than the left​ line, the distribution is skewed right. If the median is to the right of the center of the box or the left line is substantially longer than the right​ line, the distribution is skewed left. Notice that the horizontal lines are approximately equal in length and the median is in the center of the box.​ Thus, the shape of the distribution is roughly symmetric. A horizontal boxplot is above a number line from 40 to 70 in increments of 1 and consists of a box extending from 48 to 61 with a vertical line segment through the box at 55 and two horizontal line segments extending from the left and right sides of the box to 41 and 69, respectively.

3.1 Example: A professor has recorded exam grades for 15 students in his​ class, but one of the grades is no longer readable. If the mean score on the exam was 83 and the mean of the 14 readable scores is 87​, what is the value of the unreadable​ score?

The mean of a variable is computed by determining the sum of all the values of the variable in the data set and dividing by the number of observations. If x1​, x2​,..., xn are n observations of a variable from a​ sample, then the sample​ mean, x​, is found using the formula below. x = (x1+x2+...+xn) / n The mean of the 14 readable​ scores, xR​, is given by the formula below. xR= (x1+x2+...+x14) / 14 Substitute the known values into the formula for xR and solve for the sum of the 14 readable values. 87 = xR 87•14 = x1+x2+...+x14 1218 = x1+x2+...+x14 The mean of the 15 total​ scores, xT​, is given by the formula below. xT= (x1+x2+...+x14+x15) / 15 Substitute the known values into the formula for xT and solve for the sum of the 15 total scores. 83 = (x1+x2+...+x14+x15) / 15 83•15 = x1+x2+...+x15 1245 = x1+x2+...+x15 The value of the unreadable score is the difference in the two sums. ​Therefore, the value of the unreadable score is 1245−1218=27.

3.1 The following data represent the pulse rates​ (beats per​ minute) of nine students enrolled in a statistics course. Treat the nine students as a population. Complete parts ​(a) through​ (c). Student:Pulse Perpectual Bempah: 89 Megan Brooks: 74 Jeff Honeycutt: 68 Clarice Jefferson: 87 Crystal Kurtenbach: 76 Janette Lantka: 64 Kevin McCarthy: 66 Tammy Ohm: 89 Kathy Wojdya: 70 ​(a) Determine the population mean pulse. The population mean pulse is approximately _____ beats per minute. ​(Round to one decimal place as​ needed.) (b) Determine the sample mean pulse of the following two simple random samples of size 3. Sample​ 1: {Kathy, Kevin, Tammy} Sample​ 2: {Perpectual, Crystal, Clarice} The mean pulse of sample 1 is approximately _____ beats per minute. ​(Round to one decimal place as​ needed.) (c) Determine if the means of samples 1 and 2​ overestimate, underestimate, or are equal to the population mean.

(a) The population mean pulse is approximately 75.9 beats per minute. (b)The mean pulse of sample 1 is approximately 75 beats per minute. ​(Round to one decimal place as​ needed.) The mean pulse of sample​ 2, is approximately 84 beats per minute. ​(Round to one decimal place as​ needed.) (c) The mean pulse rate of sample 1 underestimates the population mean. The mean pulse rate of sample 2 overestimates the population mean. Note: Compare the mean pulse rate of sample 1 to that of the population mean to determine if the sample​ overestimates, underestimates, or is equal to the population mean.

3.5 The following data represent the dividend yields​ (in percent) of a random sample of 28 publicly traded stocks. Complete parts ​(a) to ​(c). 0.13 0.19 2.93 2.57 0.21 2.92 1.53 1.15 0.4 0.96 0 2.68 3.13 0.19 2.04 3.6 0.43 0.69 2.82 3.09 0 0.21 0 2.59 0.13 2.11 1.37 1.42 ​(a) Compute the​ five-number summary. ​(Round to two decimal places as needed. Use ascending​ order.) ​(b) Draw a boxplot of the data. A. A horizontal boxplot is above a number line from less than 0 to 4 in increments of 1 and consists of a shaded box extending from 0 to 2.6 with a vertical line segment through the box at 0.5 and a horizontal line segment extending from the right side of the box to 3.6. All values are approximate. B. A horizontal boxplot is above a number line from less than 0 to 4 in increments of 1 and consists of a shaded box extending from 0.7 to 3.1 with a vertical line segment through the box at 1.8 and two horizontal line segments extending from the left and right sides of the box to 0 and 3.6, respectively. All values are approximate. C. A horizontal boxplot is above a number line from less than 0 to 4 in increments of 1 and consists of a shaded box extending from 0.2 to 2.6 with a vertical line segment through the box at 1.3 and two horizontal line segments extending from the left and right sides of the box to 0 and 3.6, respectively. All values are approximate. ​(c) Determine the shape of the distribution from the boxplot. A. The distribution is skewed to the left. B. The distribution is skewed to the right. C. The distribution is roughly symmetric. D. The shape of the distribution cannot be determined from the boxplot.

(a) The​ five-number summary is 0​, 0.2​, 1.26​, 2.64​, 3.6. (b) C. A horizontal boxplot is above a number line from less than 0 to 4 in increments of 1 and consists of a shaded box extending from 0.2 to 2.6 with a vertical line segment through the box at 1.3 and two horizontal line segments extending from the left and right sides of the box to 0 and 3.6, respectively. All values are approximate. (c) B. The distribution is skewed to the right.

3.2 Each of the following three data sets represents the IQ scores of a random sample of adults. IQ scores are known to have a mean and median of 100. For each data​ set, determine the sample standard deviation. Then recompute the sample standard deviation assuming that the individual whose IQ is 105 is accidentally recorded as 150. For each sample​ size, state what happens to the standard deviation. Comment on the role that the number of observations plays in resistance. Click the icon to view the three data sets. For each data​ set, compute the standard deviation. 1. What is the standard deviation of the sample of size​ 5? ​(Type an integer or decimal rounded to one decimal place as​ needed.) 2. What is the standard deviation of the sample of size​ 12? ​(Type an integer or decimal rounded to one decimal place as​ needed.) 3. What is the standard deviation of the sample of size​ 30? ​(Type an integer or decimal rounded to one decimal place as​ needed.) 4. For each data set recalculate the standard​ deviation, assuming that the individual whose IQ is 105 is accidently recorded as 150. What is the standard deviation of the new sample of size​ 5? ​(Type an integer or decimal rounded to one decimal place as​ needed.) 5. What is the standard deviation of the new sample of size​ 12? ​(Type an integer or decimal rounded to one decimal place as​ needed.) 6. What is the standard deviation of the new sample of size​ 30? ​(Type an integer or decimal rounded to one decimal place as​ needed.) 7. For each sample​ size, the standard deviation increased. Comment on the role that the number of observations plays in resistance. A. As the sample size​ increases, the impact of the misrecorded data on the standard deviation remains the same. B. As the sample size​ increases, the impact of the misrecorded data on the standard deviation increases. C. As the sample size​ increases, the impact of the misrecorded data on the standard deviation decreases.

1. 6.4 ​(Type an integer or decimal rounded to one decimal place as​ needed.) 2. 7.3 ​(Type an integer or decimal rounded to one decimal place as​ needed.) 3. 9.2 ​(Type an integer or decimal rounded to one decimal place as​ needed.) 4. 20 ​(Type an integer or decimal rounded to one decimal place as​ needed.) 5. 14.7 ​(Type an integer or decimal rounded to one decimal place as​ needed.) 6. 12.2 ​(Type an integer or decimal rounded to one decimal place as​ needed.) 7. C. As the sample size​ increases, the impact of the misrecorded data on the standard deviation decreases.

3.1 A concrete mix is designed to withstand 3000 pounds per square inch​ (psi) of pressure. The following data represent the strength of nine randomly selected casts​ (in psi). 3950​, 4080​, 3100​, 3100​, 2930​, 3840​, 4090​, 4050, 3530 Compute the​ mean, median and mode strength of the concrete​ (in psi). 1. Compute the mean strength of the concrete. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The mean strength of the concrete is _____ psi of pressure. ​(Round to the nearest tenth as​ needed.) B. The mean does not exist. 2. Compute the median strength of the concrete. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The median strength of the concrete is _____ psi of pressure. ​(Round to the nearest tenth as​ needed.) B. The median does not exist. 3. Compute the mode strength of the concrete. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The mode of the strengths of the concrete is _____ psi of pressure. ​(Round to the nearest tenth as​ needed.) B. The mode does not exist.

1. a. 3630 psi of pressure. 2. a.3840 psi of pressure. 3. a.3100 psi of pressure.

The following data represent the miles per gallon for a particular make and model car for six randomly selected vehicles. Compute the​ mean, median, and mode miles per gallon. 22.6​, 25.1​, 31.7​, 27.7​, 22.9​, 39.6 1. Compute the mean miles per gallon. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The mean mileage per gallon is_______ ​(Round to two decimal places as​ needed.) B. The mean does not exist. 2. Compute the median miles per gallon. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The median mileage per gallon is ________ ​(Round to two decimal places as​ needed.) B. The median does not exist. 3. Compute the mode miles per gallon. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The mode is _____ ​(Round to two decimal places as needed. Use a comma to separate answers as​ needed.) B. The mode does not exist.

1. a. The mean mileage per gallon is 28.27 2. a. The median mileage per gallon is 26.4 3. B. no mode

3.2 Find the sample variance and standard deviation. 23​, 16​, 3​, 8​, 11 1. Choose the correct answer below. Fill in the answer box to complete your choice. ​(Type an integer or a decimal. Round to one decimal place as​ needed.) A. s^2=___ B. σ^2=_____ 2. Choose the correct answer below. Fill in the answer box to complete your choice. ​(Round to one decimal place as​ needed.) A. s=____ B. σ=______

1. s^2=58.7 2. s=7.7

3.4 A manufacturer of bolts has a​ quality-control policy that requires it to destroy any bolts that are more than 2 standard deviations from the mean. The​ quality-control engineer knows that the bolts coming off the assembly line have mean length of 7 cm with a standard deviation of 0.10 cm. For what lengths will a bolt be​ destroyed? Select the correct choice below and fill in the answer​ box(es) to complete your choice. ​(Round to one decimal place as​ needed.) A. A bolt will be destroyed if the length is greater than ____ cm. B. A bolt will be destroyed if the length is less than_____ cm or greater than _____ cm. C. A bolt will be destroyed if the length is less than ____ cm. D. A bolt will be destroyed if the length is between _____ cm and _____cm.

B. A bolt will be destroyed if the length is less than 6.8 cm or greater than 7.2 cm. Note: Create an inequality or set of inequalities to represent the length of a bolt that will be destroyed using the formula for the population​ z-score given​ below, where x is a data​ value, μ is the population​ mean, and σ is the population standard deviation. Substitute the given values into the inequality and solve for x.

3.4 Suppose babies born after a gestation period of 32 to 35 weeks have a mean weight of 2500 grams and a standard deviation of 800 grams while babies born after a gestation period of 40 weeks have a mean weight of 3000 grams and a standard deviation of 375 grams. If a 35​-week gestation period baby weighs 2125 grams and a 40​-week gestation period baby weighs 2625 ​grams, find the corresponding​ z-scores. Which baby weighs less relative to the gestation​ period? Find the corresponding​ z-scores. Which baby weighs relatively less​? Select the correct choice below and fill in the answer boxes to complete your choice. ​(Round to two decimal places as​ needed.) A. The baby born in week 40 weighs relatively less since its​ z-score,___, is larger than the​ z-score of ___r the baby born in week 35. B. The baby born in week 35 weighs relatively less since its​ z-score, ____​, is larger than the​ z-score of ____ for the baby born in week 40. C. The baby born in week 40 weighs relatively less since its​ z-score, ______​, is smaller than the​ z-score of _____ for the baby born in week 35. D. The baby born in week 35 weighs relatively less since its​ z-score, ____ is smaller than the​ z-score of _____ for the baby born in week 40.

C. The baby born in week 40 weighs relatively less since its​ z-score, −1​, is smaller than the​ z-score of −0.47 for the baby born in week 35.

3.1 Example A concrete mix is designed to withstand 3000 pounds per square inch​ (psi) of pressure. The following data represent the strength of nine randomly selected casts​ (in psi). 3975​, 4085​, 3100​, 3200​, 2950​, 3840​, 4085​, 4020, 3415 Compute the​ mean, median and mode strength of the concrete​ (in psi).

The arithmetic mean of a variable is computed by determining the sum of all the values of the variable in the data​ set, divided by the number of observations. In order to calculate the mean start by summing the strengths of the concrete. 3975+4085+3100+3200+2950+3840+4085+4020+3415=32,670 Now determine the number of​ observations, n, in the sample. There are n=9 observations. Find the mean by dividing the sum by the number of observations. x = 32,670 / 9 = 3630 psi ​Thus, the mean strength of the concrete is 3630 psi of pressure. The median of a variable is the value that lies in the middle of the data when arranged in ascending order. That​ is, half the data are below the median and half the data are above the median. We use M to represent the median. To find the median start by arranging the data in ascending order. 2950​, 3100​, 3200​, 3415​, 3840​, 3975​, 4020​, 4085​, 4085 Since the data set has an odd number of​ values, the median is the data value that is exactly in the middle of the data set. M=3840 psi ​Thus, the median strength of the concrete is 3840 psi of pressure. The mode of a variable is the most frequent observation of the variable that occurs in the data set. To determine the​ mode, start by determining the frequency of each strength. Since 4085 psi occurs more than once in the data​ set, it has a frequency greater then 1. Determine the mode of the strengths. The mode is the psi that has the greatest frequency. Mode=4085 psi ​Thus, the mode of the strengths of the concrete is 4085 psi of pressure.

3.3 example ​Recently, a random sample of 25-34 year olds was​ asked, "How much do you currently have in​ savings, not including retirement​ savings?" The data in the table represent the responses to the survey. Approximate the mean and standard deviation amount of savings. Click the icon to view the frequency distribution for the amount of savings.​

The sample mean is given by the formula​ below, where xi is the midpoint or value of the ith​ class, fi is the frequency of the ith​ class, and n is the number of classes. x=∑xifi / ∑fi = (x1f1+x2f2+...+xnfn) / f1+f2+...+fn While either the formulas or technology can be​ used, for this​ exercise, use technology. First determine the class midpoint of each class by adding consecutive lower class limits and dividing the result by 2. Begin with the lowest class. $0+$200 / 2=​$100 Use the same process to calculate the midpoints for the rest of the classes. Class: Class​ Midpoint, xi ​$0-$199 :$100 ​$200-​$399: $300 ​$400-$599: $500 ​$600-$799: $700 ​$800-$999: $900 ​$1000-​$1199: $1100 ​$1200-$1399:$1300 Now use technology to calculate the sample​ mean, rounding to the nearest whole number. The sample mean for the amount of savings is ​$250. The sample standard deviation is given by the formula​ below, where xi is the midpoint or value of the ith​ class, fi is the frequency of the ith​ class, and x is the sample mean. s=sqrt (∑xi−x2fi) / ∑fi−1 Use technology to calculate the sample standard​ deviation, rounding to the nearest whole number. s=236 ​Therefore, the sample standard deviation is ​$236.

3.4 example Suppose babies born after a gestation period of 32 to 35 weeks have a mean weight of 2400 grams and a standard deviation of 700 grams while babies born after a gestation period of 40 weeks have a mean weight of 2600 grams and a standard deviation of 410 grams. If a 33​-week gestation period baby weighs 1850 grams and a 40​-week gestation period baby weighs 2050 grams, find the corresponding​ z-scores. Which baby weighs less relative to the gestation​ period?

To determine which baby weighs relatively less​, compute each​ baby's z-score. The population​ z-score can be found using the formula​ below, where x is the​ weight, μ is the appropriate population mean​ weight, and σ is the appropriate population standard deviation. z=x−μ / σ First identify each value for the formula. Identify the value of x for the baby born during week 33. x=1850 Identify the value of μ for this baby. μ=2400 Identify the value of σ for this baby. σ=700 Substitute the values into the formula and calculate the​ z-score, rounding to two decimal places. z = x−μ / σ = 1850−2400700 = −0.79 Compute the​ z-score for the baby born during week 40​, rounding to two decimal places. z = x−μ / σ = 2050−2600410 = −1.34 ​Z-scores measure the number of standard deviations an observation is above or below the mean. As the magnitude of the​ z-score increases, the relative difference of the observation from the mean increases. ​Therefore, compare the​ z-scores of the two babies. The baby with the larger​ z-score weighs relatively​ more, and the baby with the smaller​ z-score weighs relatively less.