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STATS Quizzes and chapters 1-3

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Boxplots show less detail than histograms or stemplots, so they are better for comparing distributions than displaying a distribution.

What proportion of observations under a smoothed normal curve are exactly over the point 1.5 (z = 1.5)?

0 For a standard Normal (or any Normal distribution) the chance of the variable exactly equaling any value is 0. This is a consequence of smoothing histograms from data. We are interested in exactly 1.5 as the value of the variable.

True or False: Using the standard Normal table (use Table A in your textbook, or a graphing calculator), the area to the right of z = 0.94 is 0.1736.

True The table (or a graphing calculator) gives the area to the left of 0.94 as 0.8264. The area to the right will be 1 - 0.8264 = 0.1736. Note that a z-score of -0.94 would also give the same result because the graph is symmetric.

True or false: No single numerical measure of variability adequately describes the skewed distributions.

True: Because the two sides of a skewed distribution have different variabilities, a single measure such as interquartile range is not useful for describing its variability.

True or false: The mean can be negative.

True: If the data contain negative values, their sum can be negative and the mean can be negative.

True or false: The median measures the center of the data set.

True: Mean and median are both measures of center.

five number summary includes

is better than mean and standard deviation for describing skewed distribution.

Thinking of density curve shapes, if an instructor gave a test and most students did well, the distribution of grades should be ________ skewed.

left If most students do well on the test, the peak would be on the high end of the grade distribution; there would most likely be a long left hand tail.

For a small quantitative data set, like a small class's exam scores, one should use a

stemplot.

Too much cholesterol in the blood increases the risk of heart disease. The cholesterol levels of young women aged 20 to 34 years vary approximately normally with a mean of 185 milligrams per deciliter (mg/dl) and a standard deviation of 39 mg/dl. Cholesterol levels for middle-aged men vary normally with a mean of 222 mg/dl and a standard deviation of 37 mg/dl. Sandy is a young woman with a cholesterol level of 220. Her father has a cholesterol level of 250. Who has relatively higher cholesterol?

Sandy The z-score for Sandy is The z-score for Sandy's father is Since Sandy's z-score is larger, her cholesterol level is slightly higher. Text reference: section 3.5.

Which one of the following statements about the standard deviation is TRUE?

Standard deviation is not resistant to outliers; removing outliers reduces standard deviation.

3. The distribution of bladder volume in men is approximately Normal with mean 550 ml and standard deviation 100 ml. What percent of men have a bladder volume smaller than 450 ml? Round you answer to the nearest whole number.

(16 ans) 450 ml is one standard deviation below the mean. We expect about 68% of all men to have bladder volumes within one standard deviation of 550. The remaining 32% is split in half because of symmetry.

The formula for calculating degrees of freedom of the variance or the standard deviation is n _____ 1.

True or false: σ represents a sample standard deviation.

False

True or False: For any Normal distribution, the two z-scores that divide the area into the middle 99.7% are z = ±2.75.

False If 99.7% is in the middle of the distribution, there is 0.0015 (0.15%) in either tail. Read the Standard Normal table to find that z = 2.97.

True or False: If the area to the right of a z-score is less than 0.5, the z-score is negative.

False If the area to the right is less than 0.5, the value is above the mean.

True or False: The area under a density curve depends on the shape of the curve.

False The area under a density curve is always 1, or 100%.

True or false: The median has the same unit of measure as the data values.

True

True or false: You should always examine your data for outliers.

True

A shopper at a local supermarket spent the following amounts in his last eight trips to the store: $32.92 $14.14 $30.80 $27.34 $28.34 $75.58 $27.42 $22.94 The value of the median is _________________. Give answer as $XX.XX.

$27.88

f a z-score is negative, the area to its right is greater than 0.5.

True

It appears that people who are mildly obese are less active than leaner people. One study looked at the average number of minutes per day that people spend standing or walking. Among mildly obese people, minutes of activity is normally distributed with mean 373 minutes and standard deviation of 67 minutes. What proportion of mildly obese individuals spend between 350 and 380 minutes per day standing or walking?

0.173 350 minutes standardizes to z = -0.34; 380 minutes standardizes to z = 0.10. Using the Normal table, the area to the left of z = -0.34 is 0.3669. the area to the left of z = 0.10 is 0.5398. Subtracting, we have 0.5398 - 0.3669 = 0.1729.

T. The lifetime of a 2-volt nonrechargeable battery in constant use has a Normal distribution with a mean of 516 hours and a standard deviation of 20 hours. The proportion of batteries with lifetimes exceeding 520 hours is approximately

0.4207. The z-score corresponding to 520 is (520 - 516)/20 = 0.20. Based on the standard Normal table (Table A), the area to the right is 1 - 0.5793 = 0.4207, which corresponds to the proportion with lifetimes exceeding 520. Text reference: sections 3.5 to 3.7.

Use the standard Normal table (use the table reader, the table in your textbook, or a graphing calculator) to find the area between the following two z-scores: -1.98 and 0.50. Round your answer to four decimal places

0.6676 The area to the left of 0.50 is 0.6915. The area to the left of -1.98 is 0.0239. For the area between the two, subtract 0.6915 - 0.0239 = 0.6676. On the graphing calculator, use Normalcdf(-1.89,0.5,0,1)

To calculate the quartiles:

1. Arrange the observations in increasing order and locate the median, M, in the ordered list of observations. 2. The first quartile, Q1, is the median of the observations whose position in the ordered list is to the left of the location of the overall median. 3. The third quartile, Q3, is the median of the observations whose position in the ordered list is to the right of the location of the overall median.

The machine scored parts of the SAT exam (critical reading and mathematics) are designed so that scores approximately follow the model below. The standard deviation of this distribution is _______. 200, 300, 400, 500, 600, 700, 800

100

The most common intelligence quotient (IQ) scale is normally distributed with a mean of 100 and a standard deviation of 15. Many school districts across the country seek to identify "gifted and talented" children for special enrichment programs. Typically, these children must have IQ scores in the top 5%. What is the minimum score to qualify a child for these programs?

125 Based on the standard Normal table (Table A), a z-score of 1.645 (halfway between z = 1.64 with 0.9495 and z = 1.65 with 0.9505 below that z-score) will have 95% of all students below that value, so 5% above it (1.645)(15) + 100 = 124.675, so the score rounds to 125. Text reference: section 3.8.

The tallest person ever confirmed by the Guinness records organization was Robert Wadlow of the United States, at 8 feet 11.1 inches tall. Heights of American men are approximately Normal with mean 69.3 inches and standard deviation 2.8 inches. What is Wadlow's z-score? Round your answer to one decimal place.

13.5 Converted to inches, he was 8*12+11.1 = 107.1 inches tall. His z-score is (107.1 - 69.3)/2.8 = 13.5.

Too much cholesterol in the blood increases the risk of heart disease. The cholesterol levels of young women aged 20 to 34 years vary approximately normally with a mean of 185 milligrams per deciliter (mg/dl) and a standard deviation of 39 mg/dl. Approximately what percent of young women in this age group will have cholesterol levels of less than 150 mg/dl?

18.5% The z-score for a cholesterol level of 150 is From the standard Normal table (Table A), the area below z = -0.90 is 0.1841 (software gives 0.1847 from a more exact z-score). Text reference: sections 3.5 to 3.7.

High blood cholesterol increases your risk of heart attack and stroke. Cholesterol levels in young women are approximately Normal with mean 189 mg/dl and standard deviation 40 mg/dl. About 34% of women will have levels between _______________.

189 and 229 In a Normal distribution, about 68% of all observations are within one standard deviation of the mean. 229 is one standard deviation above the mean and 189 is the mean. Half of the 68% will be between 189 and 229.

A university wants to select individuals for their honors program partly on the basis of SAT scores. Total scores for the three parts of the test are approximately normally distributed with mean 1500 and standard deviation 250. If they want only the top 5% to qualify, what total SAT score must they equal or exceed? SAT scores are reported in multiples of 10.

1920 The z-score with 95% of area under the curve to its left is 1.645. SAT = 1.645*250 + 1500 = 1911.25, so they must score at least 1920 in multiples of 10.

According to the 68-95-99.7 Rule, approximately 99.7% of the area in any Normal distribution is within 3 standard deviations of the mean μ. The actual number z of standard deviations for 99.7% of all observations to be within z σ of μ is _______.

2.97 If 99.7% is in the center of the distribution, there is 0.0015 (0.15%) in either tail. Read the Standard Normal table to find that z = 2.97.

A good use of z-scores is to compare values in two different distributions. Suppose your instructor will drop the low test when computing final grades, but he curves grades on each test. You had a 92 on the first test, and an 85 on the second. The first test had a mean of 83 and standard deviation of 6. The second test had a mean of 75 and standard deviation 3. The z-score for your second test is ______. Round your answer to two decimal places.

3.33 The z-score is (85 - 75)/3 = 3.33.

True or False: If a z-score is negative, the area to its right is greater than 0.5.

True If a z-score is negative, the value is below the mean. The area to the right will be more than 0.5.

An instructor in a large lecture class found at the end of the semester that the total point distribution in his class was approximately normal with a mean of 530 and a standard deviation of 80. About what percent scored between 490 and 590?

46.5%. If using the standard Normal table (Table A), first find the z-scores for 490 and 590. . For x = 490, this is -0.50. For x = 590, this is 0.75. We find the proportion scoring 590 or less and subtract the proportion scoring 490 or less. This gives 0.7734 - 0.3085 = 0.4649. Software gives 0.4648. Text reference: section 3.7.

2) ____% of the data are less than the median.

50%

___________% of the data are less than the median.

50%

The lifetime of a 2-volt nonrechargeable battery in constant use has a normal distribution with a mean of 516 hours and a standard deviation of 20 hours. Ninety percent of all batteries have a lifetime of less than

541.60 hours. In the body of the standard Normal table (Table A), we find the z-value corresponding to the proportion 0.9000 is 1.28. We need to use the formula , where z = 1.28, to get the correct answer, or x = 516 + (1.28)(20) = 541.6. Text reference: section 3.8.

632 If the top 10% of students will get an A, 90% of them will score lower. In the body of the standard Normal table (Table A), we find the z-value corresponding to the proportion 0.9000 is 1.28. We need to use the formula , where z = 1.28, to get the correct answer, or x = 530+1.28*80 = 632.4. Text reference: section 3.8.

Attendance at a university's basketball games follows a Normal distribution with mean μ = 8,000 and standard deviation σ = 1,000. 60% of all games will have more than _______ people in attendance.

7750 From the z-score table (Table A in the back of the book), the z-score corresponding to 60% above (40% below) is -0.25. Solving for X, the number of people, X = -0.25*1000 + 8000 = 7750. If using the graphing calculator, use invNorm(0.4,8000,1000)

Let X be a random variable whose distribution is Normal with a mean of 100 and a standard deviation of 10. The proportion of observations above 115 is equal to the proportion below _____. (Enter your answer as a number.)

85 The proportion more than 15 units below the mean is the same as the proportion more than 15 units above the mean, because of symmetry in Normal distributions.

Hemoglobin is the compound in red blood cells that carries oxygen to the body. The distribution of hemoglobin in women in g/dl of blood is approximately normally distributed with mean 14 and standard deviation 1. According to MedicineNet.com, "normal" hemoglobin levels range from 12 to 16. What percent of women have normal hemoglobin levels?

95 12 and 16 are each a distance 2 standard deviations away from the mean. Normal distributions have approximately 95% of their area within 2 standard deviations of the mean.

95% The 68-95-99.7 rule says that approximately 95% will be within two standard deviations of the mean. This is 530 - 2*80 = 370 to 530 + 2*80 = 690. Text reference: section 3.4 or sections 3.5 to 3.7.

Order the datasets from smallest to largest in terms of standard deviation.

Correct! Because Dataset A has more data concentrated in the center nearest the mean, it has the smallest standard deviation. Dataset B has the most observations farthest from the mean so it has the largest standard deviation.

True or false: The mean and median are never equal.

False

True or false: The span of the central box in a boxplot shows the variability of almost all of the data.

False

True or false: The standard deviation is resistant to outliers.

False

True or False: Mensa is a society for "geniuses." One way to qualify for membership is having an IQ at least 2.5 standard deviations above average. IQs are approximately Normally distributed, according to the WAIS scale. More than 2% of people will qualify.

False To qualify, a person's z-score must be at least 2.5. Find the area above z = 2.5 by subtracting the table entry (0.9938) from 1. This turns out to be 0.0062, which is less than 1%.

Z-scores measure the proportion of observations less than a particular value.

False Z-scores do not measure area under a Normal distribution; they measure position in a distribution. If the distribution is Normal, once you have the z-score, you can find the proportion less than the value of interest using the standard Normal table.

The central box in a boxplot shows the distance between:

First quartile and third quartile

Which measures should be used to describe a very right skewed distribution?

Five-number summary

The value of the interquartile range is _____. Give your answer rounded to one decimal place.

Interquartile range is IQR = 128.5 - 86.0 = 42.5.

Which one of the following is NOT part of the five-number summary?

Mean

Which measures should be used to describe an approximately symmetric distribution with no outliers?

Mean and standard deviation can be used for a reasonably symmetric distribution with no outliers.

There are eight boys in a pre-school class. Their mean height is 33 inches and their median height is 33 inches. The tallest boy whose height is 38 inches moves away and is replaced by a boy whose height is 39 inches. How does this affect the mean?

Mean will increase.

Missy took the ACT and was told her standard score (z-score) is -1. Frank took the SAT and was told his standard score (z-score) is -2. Which student has a better chance of getting admitted to college based on their test score?

Missy Missy's z-score is higher than Frank's (hers is 1 standard deviation below average while Frank's is 2 standard deviations below average), so she performed better on the standardized test than Frank did. Text reference: section 3.5.

A good use of z-scores is to compare values in two different distributions. Suppose your instructor will drop the low test when computing final grades, but he curves grades on each test. You had a 92 on the first test, and an 85 on the second. The first test had a mean of 83 and standard deviation of 6. The second test had a mean of 75 and standard deviation 3. The test with the relatively better score is ____________.

Test 2 Compute a z-score for each test. The first is (92 - 83)/6 = 1.5. The second is (85 - 75)/3 = 3.333.

A shopper at a local supermarket spent the following amounts in her last eight trips to the store: $32.92 $14.14 $30.80 $28.34 $75.58 $36.33 $33.51 $22.94 The amount spent of _________ is most likely an outlier. Give answer as $XX.XX.

The amount spent of $75.58 is considerably more than any of the other amounts.

A shopper at a local supermarket spent the following amounts in his last eight trips to the store: $32.92 $14.14 $30.80 $28.34 $75.58 $36.33 $33.51 $22.94 The mean of these data is $34.32. If the amount of $75.58 were a typo and should be $25.58, what would happen to the mean?

The mean would decrease. Since $25.58 would be added instead of $75.58, the sum would be smaller and the mean would be smaller.

graphing calculator) to find the area to the right of z = 0.94

The table (or a graphing calculator) gives the area to the left of 0.94 as 0.8264. The area to the right will be 1 - 0.8264 = 0.1736.

The z-score is (85 - 75)/3 = 3.33.

Mean and standard deviation

This distribution is approximately symmetric with no outliers so the mean and standard deviation can be used to describe these data.

Hemoglobin is the compound in red blood cells that carries oxygen to the body. The distribution of hemoglobin in women in g/dl of blood is approximately normally distributed with mean 14 and standard deviation 1. Too little hemoglobin, and you're anemic. Too much, and (unless you live at high altitudes), you can have other problems. True or False: 18.5% of women will have hemoglobin levels below 12 (be anemic) or above 15. Use the 68-95-99.7% rule to solve.

True

The standard deviation has the same unit of measure as the original data.

True

True or false: Boxplots are best used for side-by-side comparisons of two or more distributions.

True

True or false: Even though two datasets have the same values for the mean and standard deviation, their shapes can still be different.

True

When computing the sample variance using a sample of size n = 84, we divide the sum of squared deviations by ______. Give answer as a whole number.

Variance is computed by dividing by n - 1 = 84 - 1 = 83.

The bars on a histogram

always touch one another.

When ordering vinyl replacement windows, the following variables are specified for each window. Which of these variables is quantitative?

area of the window opening in square inches

A survey of college students collected information on several variables: Distance from Home, Age, Major, Gender, and Class. The variable Class is

categorical.

True or false? A pie chart and bar graph can be used interchangeably as a graphical display.

false

A woman is told her weight has a standard score (z-score) of -1.5. This means that

her weight is 1.5 standard deviations below average

An instructor asked students to name their favorite color. An appropriate graph to display the data is a

pie chart

You will find the ______ of any Normal distribution 0.675 standard deviations from the mean.

quartiles 0.7500 is located in the table between 0.7486 and 0.7517. These correspond to z = 0.67 and 0.68. So, there is 75% of the area below z = 0.675. That makes it Q3. Similarly, there is 25% of the area below z = -0.675.

The symbol representing the sample standard deviation is _________.

The mean is a measure of center whereas the standard deviation measures the ____________ of data about the mean.

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