Statistics
The accompanying 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). (a) Compute the population standard deviation. (b) Determine the sample standard deviation of the following three simple random samples of size 3. Sample 1 (Kathy,T,P) 2 (MCT) 3 (Kevin, T,J) (c) Which samples underestimate the population standard deviation? Which overestimate the population standard deviation?
(a) 8.2 (b) Sample 1 = 5.5, sample 2 = 6.7, sample 3 = 6.8 (c) All
(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. (b) The five-number summary is
(a) Skewed left (b) 0,14,17,19,20
Which histogram depicts a higher standard deviation?
Histogram a depicts the higher standard deviation, because the distribution has more dispersion. KEY NOTE: pay close attention to the values on the bottom or sides of the chart.
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?
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. When data are either skewed left or skewed right, there are extreme values in thetail, 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.
The following data represent the weights (in grams) of a random sample of 50 candies. (a) Determine the sample standard deviation weight. (b) On the basis of the histogram on the right, comment on the appropriateness of using the empirical rule to make any general statements about the weights of the candies. (c) (c) Use the empirical rule to determine the percentage of candies with weights between 0.7 and 0.98 gram. Hint: x=0.84. (d) Determine the actual percentage of candies that weigh between 0.7 and 0.98 gram, inclusive. To calculate the percentage, count all the candies that weigh between 0.7 and 0.98 gram, inclusive, divide this number by 50, and multiply by 100. (e) Use the empirical rule to determine the percentage of candies with weights more than 0.91 gram. (f) Determine the actual percentage of candies that weigh more than 0.91 gram. To calculate the percentage, count all the candies that weigh more than 0.91 gram, divide this number by 50, and multiply by 100.
(a) 0.07 (b) The histogram is bell-shaped so the empirical rule can be used. (c)95% (d)98% (e) 16% (f) 18%
Another measure of central tendency is the trimmed mean. It is computed by determining the mean of a data set after deleting the smallest and largest observed values. Compute the trimmed mean for the data given in the accompanying table. Is the trimmed mean resistant to changes in the extreme values in the given data? 0.81 0.88 0.82 0.90 0.90 0.84 0.84 0.91 0.94 0.86 0.86 0.86 0.88 0.87 0.89 0.91 0.86 0.87 0.93 0.88 0.83 0.98 0.87 0.93 0.91 0.85 0.91 0.91 0.86 0.89 0.87 0.81 0.88 0.88 0.89 0.76 0.82 0.83 0.90 0.88 0.84 0.93 0.81 0.90 0.88 0.92 0.85 0.84 0.84 0.86 (a) What is the trimmed mean? (b) Is the trimmed mean resistant to changes in the extreme values for the given data?
(a) 0.8728=0.873 (b) Yes, because changing the extreme values does not change the trimmed mean.
Violent crimes include rape, robbery, assault, and homicide. The following is a summary of the violent-crime rate (violent crimes per 100,000 population) for all states of a country in a certain year. Complete parts (a) through (d). Q1=272.8, Q2=388.5, Q3=529.7 (a) Provide an interpretation of these results. Choose the correct answer below. (b) Determine and interpret the interquartile range. Interpret the interquartile range. Choose the correct answer below. (c) The violent-crime rate in a certain state of the country in that year was 1,459. Would this be an outlier? (d) Do you believe that the distribution of violent-crime rates is skewed or symmetric?
(a) 25% of the states have a violent-crime rate that is 272.8 crimes per 100,000 population or less. 50% of the states have a violent-crime rate that is 388.5 crimes per 100,000 population or less. 75% of the states have a violent-crime rate that is 529.7 crimes per 100,000 population or less. (b) Q3-Q1 = 256.9 The middle 50% of all observations have a range of 256.9 crimes per 100,000 population. (c) Yes, because it is greater than the upper fence. Upper Fence = 915.05 Lower Fence = -112.55 (d) The distribution of violent-crime rates is skewed right.
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) to (c). 62,62,63,75,76,83,84,85,89 (a) Determine the population mean pulse. (b) Determine the sample mean pulse of the following two simple random samples of size 3. Sample 1: {Crystal, Megan, Tammy} Sample 2: {Perpectual, Janette, Crystal} (c) Determine if the means of samples 1 and 2 overestimate, underestimate, or are equal to the population mean.
(a) 679/9=75 (b) sample 1 = 74 & sample 2 = 71 (c) The mean pulse rate of sample 1 & 2 "Underestimates" the population mean.
At one point the average price of regular unleaded gasoline was $3.39 per gallon. Assume that the standard deviation price per gallon is $0.07 per gallon and use Chebyshev's inequality to answer the following. (a) What percentage of gasoline stations had prices within 3 standard deviations of the mean? (b) What percentage of gasoline stations had prices within 1.5 standard deviations of the mean? What are the gasoline prices that are within 1.5 standard deviations of the mean? (c) What is the minimum percentage of gasoline stations that had prices between $3.11 and $3.67?
(a) At least 88.89% of gasoline stations had prices within 3 standard deviations of the mean. (b) At least 55.56% of gasoline stations had prices within 1.5 standard deviations of the mean. The gasoline prices that are within 1.5 standard deviations of the mean are $3.285 to $3.495 (c) 93.75% is the minimum percentage of gasoline stations that had prices between $3.11 and $3.67.
The following data represent the weights (in grams) of a simple random sample of a candy. 0.81, 0.86, 0.86, 0.89, 0.90, 0.90, 0.92, 0.92, 0.93, 0.93 Determine the shape of the distribution of weights of the candies by drawing a frequency histogram and computing the mean and the median. Which measure of central tendency best describes the weight of the candy? (a) choose the correct histogram (b) Is the histogram for the data set skewed right, skewed left, or symmetric? (c) The mean weight of the candies is ________ grams? (d) The median weight of the candies is _______ grams? (e) Which measure of central tendency best describes the weight of the candy?
(a) Bar graph skewed left with a gap before the lowest decimal. (b) Skewed left (c) 0.892 (d) 0.90 (e) median
A sample of 20 registered voters was surveyed in which the respondents were asked, "Do you think Chang, Johnson, Ohm, or Smith is most qualified to be asenator?" The results of the survey are shown in the table. Smith Johnson Chang Ohm Smith Ohm Johnson Smith Johnson Ohm Johnson Johnson Johnson Ohm Chang Smith johnson Ohm Smith Ohm (a) Determine the mode candidate. (b) Do you think it would be a good idea to rotate the candidate choices in the question? Why?
(a) Johnson (b) Yes, to avoid response bias
Find the sample variance and standard deviation. 23,13,4,8,9 (a) s* or a* (b) a or s
(x) = add all values of x or n = 54(xi) (xi*) = add all values of x* or n* =730 730-(54)*/5 over 5-1 = 730-2916/5 over 5-1 730-583.2/5-1 = 146.8/4=36.7 (a) s*=36.7 (b) s* = 6
A professor has recorded exam grades for 10 students in his class, but one of the grades is no longer readable. If the mean score on the exam was 81 and the mean of the 9 readable scores is 86, what is the value of the unreadable score?
9.86=774 10.81=810 810-774 = 36
For the histogram on the right determine whether the mean is greater than, less than, or approximately equal to the median. Justify your answer. 3,4,4,4,5,5,7,7,7,7,8,10
Mean - 3,4,4,4,5,5,7,7,7,7,8,10 = 12 numbers value of 71 71/12=5.9166 Median - 5 Answer = X = M because the histogram is symmetric.
An insurance company crashed four cars of the same model at 5 miles per hour. The costs of repair for each of the four crashes were $434, $431, $492, and $246 . Compute the mean, median, and mode cost of repair.
Mean = All numbers divided by 4 = $400.75 Median = all numbers arranged from smallest to largest then add the number(s) in the middle, then divide their total value by the amount of numbers it took to get that value = 434+431/2= 432.50 Mode = None
