Homework: 3.2 Measures of Dispersion

Find the sample variance and standard deviation. 22​, 10​, 4​, 7​, 8 Choose the correct answer below. Fill in the answer box to complete your choice. ​A. s squared = ______ B. sigma squared = --------- Choose the correct answer below. Fill in the answer box to complete your choice. ​(Round to one decimal place as​ needed.) A. s = ________ B. sigma = -------

48.2, 6.9

Which histogram depicts a higher standard​ deviation?

Histogram a depicts the higher standard​ deviation, because the distribution has more dispersion.

What is a similarity between the Empirical Rule and​ Chebychev's Theorem? A. Both calculate the variance and standard deviation of a sample. B. Both apply only to symmetric and​ bell-shaped distributions. C. Both estimate proportions of the data contained within k standard deviations of the mean. D. Both do not require the data to have a sample standard deviation.

C. Both estimate proportions of the data contained within k standard deviations of the mean.

Find the population variance and standard deviation. 6​, 12​, 20​, 24​, 28

Sigma2 = 64, sigma = 8

The ages​ (in years) of a random sample of shoppers at a gaming store are shown. Determine the​ range, mean,​ variance, and standard deviation of the sample data set. 12​, 19​, 23​, 15​, 18​, 16​, 21​, 18​, 14​, 19

The range is 11. The mean is 17.5. ​The variance is 10.94. The standard deviation is 3.3.

An insurance company crashed four cars in succession at 5 miles per hour. The cost of repair for each of the four crashes was ​$422​, ​$464​, ​$416​, ​$232 . Compute the​ range, sample​ variance, and sample standard deviation cost of repair.

The range is ​$232. s2 = 10657 dollars squared ​ s = $103.23

What is meant by the phrase​ "degrees of​ freedom" as it pertains to the computation of the sample standard​ deviation? Choose the correct answer below. A. The degrees of freedom refers to the number of observations that must be taken into account when computing the sample standard​ deviation, s, in order to ensure that the sum of the deviations about the mean is equal to zero. B. There are n-1 degrees of freedom because the first n-1 observations have to be certain values in order for the nth value to have the freedom to be any value such that the sum of the deviations about the mean is equal to zero. C. There are n-1 degrees of freedom in the computation of the sample standard​ deviation, s, because it is obtained by dividing by n-1. D. There are n-1 degrees of freedom in the computation of s because an unknown​ parameter, mu​, is estimated by x. For each parameter​ estimated, 1 degree of freedom is lost.

D. There are n-1 degrees of freedom in the computation of s because an unknown​ parameter, mu​, is estimated by x. For each parameter​ estimated, 1 degree of freedom is lost.

The weight of an organ in adult males has a​ bell-shaped distribution with a mean of 325 grams and a standard deviation of 25 grams. Use the empirical rule to determine the following. ​(a) About 68​% of organs will be between what​ weights? ​(b) What percentage of organs weighs between 275 grams and 375 ​grams? ​(c) What percentage of organs weighs less than 275 grams or more than 375 ​grams? ​(d) What percentage of organs weighs between 300 grams and 375 ​grams?

(a) 300 and 350 grams ​(b) 95​% ​(c) 5​% ​ ​(d) 81.5​% ​

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 18. Use the empirical rule to determine the following. ​(a) What percentage of people has an IQ score between 82 and 118​? ​(b) What percentage of people has an IQ score less than 82 or greater than 118​? ​(c) What percentage of people has an IQ score greater than 136​?

(a) 68 (b)32 (c)2.5

At one point the average price of regular unleaded gasoline was ​$3.56 per gallon. Assume that the standard deviation price per gallon is ​$0.09 per gallon and use​ Chebyshev's inequality to answer the following. ​(a) What percentage of gasoline stations had prices within 2 standard deviations of the​ mean? ​(b) What percentage of gasoline stations had prices within 2.5 standard deviations of the​ mean? What are the gasoline prices that are within 2.5 standard deviations of the​ mean? ​(c) What is the minimum percentage of gasoline stations that had prices between ​$3.20 and ​$3.92​?

(a) 75​ ​​(b) 84​, ​3.34, ​3.79. ​(c) 93.75​

A certain standardized​ test's math scores have a​ bell-shaped distribution with a mean of 515 and a standard deviation of 105. Complete parts​ (a) through​ (c). (a) What percentage of standardized test scores is between 200 and 830​? ​(b) What percentage of standardized test scores is less than 200 or greater than 830​? ​(c) What percentage of standardized test scores is greater than 725​?

(a) 99.7 (b) .3 (c) 2.5

True or​ False: Chebyshev's inequality applies to all distributions regardless of​ shape, but the empirical rule holds only for distributions that are bell shaped. ​A. False, both​ Chebyshev's inequality and the empirical rule will only work for​ bell-shaped distributions. B. ​True, Chebyshev's inequality is less precise than the empirical​ rule, but will work for any​ distribution, while the empirical rule only works for​ bell-shaped distributions. C. ​False, the empirical rule is less precise than​ Chebyshev's inequality, but will work for any​ distribution, while​ Chebyshev's inequality only works for​ bell-shaped distributions. D. ​False, both​ Chebyshev's inequality and the empirical rule will work for any distribution.

B. ​True, Chebyshev's inequality is less precise than the empirical​ rule, but will work for any​ distribution, while the empirical rule only works for​ bell-shaped distributions.

What is a difference between the Empirical Rule and​ Chebychev's Theorem? A. ​Chebychev's Theorem applies only to distributions which are approximately symmetric or​ bell-shaped and the Empirical Theorem has no restrictions. B. ​Chebychev's Theorem estimates proportions of data contained within infinite standard deviations and the Empirical Rule has a limit of 5 standard deviations. C. The Empirical Rule assumes the distribution is aproximately symmetric and​ bell-shaped and​ Chebychev's Theorem makes no assumptions. D. The Empirical Rule assumes a small data set​ (less than 50​ values) where​ Chebychev's Theorem has no limit on data size.

C. The Empirical Rule assumes the distribution is aproximately symmetric and​ bell-shaped and​ Chebychev's Theorem makes no assumptions

What makes the range less desirable than the standard deviation as a measure of​ dispersion? Choose the correct answer below. A. The range is biased. B. The range is resistant to extreme values. C. The range does not use all the observations. D. The range describes how​ far, on​ average, each observation is from the mean.

C. The range does not use all the observations.

In a statistics​ class, the standard deviation of the heights of all students was 4.2 inches. The standard deviation of the heights of males was 3.2 inches and the standard deviation of females was 3.1 inches. Why is the standard deviation of the entire class more than the standard deviation of the males and females considered​ separately? Choose the correct answer below. A. The standard deviation of the entire class is more than the standard deviation of the males and females considered separately because the male and female sample sizes are smaller than the sample of the entire class. B. The standard deviation of the entire class is more than the standard deviation of the males and females considered separately because the smaller the​ sample, the smaller the sample standard deviation. C. The standard deviation of the entire class is more than the standard deviation of the males and females considered separately because there is more dispersion in the male and female samples than the entire class. D. The standard deviation of the entire class is more than the standard deviation of the males and females considered separately because the distribution of the entire class has more dispersion.

D. The standard deviation of the entire class is more than the standard deviation of the males and females considered separately because the distribution of the entire class has more dispersion

True or​ False: 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. A. True, because the standard deviation is the difference between the largest and smallest observation. When the standard deviation is​ larger, there is more distance between the largest and smallest​ observation, and​ therefore, more dispersion in the distribution. B. ​False, because the larger the standard deviation​ is, the less dispersion the distribution has. C. ​False, because the standard deviation measures the spread of the​ distribution, not the dispersion of the distribution. D. ​True, because the standard deviation describes how​ far, on​ average, each observation is from the typical value. A larger standard deviation means that observations are more distant from the typical​ value, and​ therefore, more dispersed.

D. ​True, because the standard deviation describes how​ far, on​ average, each observation is from the typical value. A larger standard deviation means that observations are more distant from the typical​ value, and​ therefore, more dispersed.

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