3. Describing Data Numerically

z Scores If your score on your next statistics test is converted to a z score, which of these z scores would you prefer: -2.00, -1.00, 0, 1.00, 2.00? Why?

Why Divide by ? Let a population consist of the values 9 cigarettes, 10 cigarettes, and 20 cigarettes smoked in a day (based on data from the California Health Interview Survey). Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.)

a. Find the variance of the population {9 cigarettes, 10 cigarettes, 20 cigarettes}.

Comparing Standard Deviations The standard deviation of batting averages of all teams in the American League is 0.008. The standard deviation of all players in the American League is 0.02154. Why is there less variability in team batting averages?

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.

Super Bowl Ages Listed below are the ages of the same 11 players used in the preceding exercise. How are the resulting statistics fundamentally different from those found in the preceding exercise?

41 24 30 31 32 29 25 26 26 25 30

Identical Values Compute the sample standard deviation of the following test scores: 78, 78, 78, 78. What can be said about a data set in which all the values are identical?

In Problems 5–10, by hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated.

Sample: 83, 65, 91, 87, 84

Travel Time to Work The frequency distribution listed in the table represents the travel time to work (in minutes) for a random sample of 895 U.S. adults.

b. Approximate the standard deviation travel time to work for U.S. adults.

Large Data Sets from Appendix B. In Exercises 25–28, refer to the indicated data set in Appendix B. Use software or a calculator to find the range, variance, and standard deviation. Express answers using appropriate units, such as “minutes.”

Earthquakes Use the magnitudes (Richter scale) of the 600 earthquakes listed in Data Set 24 “Earthquakes” in Appendix B. In 1989, the San Francisco Bay Area was struck with an earthquake that measured 7.0 on the Richter scale. If we add that value of 7.0 to those listed in the data set, do the measures of variation change much?

Using and Interpreting Concepts

Finding the Range of a Data Set In Exercises 9 and 10, find the range of the data set represented by the graph.

Finding the Sample Mean and Standard Deviation for Grouped Data In Exercises 39 and 40, make a frequency distribution for the data. Then use the table to find the sample mean and the sample standard deviation of the data set.

3 3 5 3 8 0 3 9 6 6 7 1 6 3 2 6 9 1 8 5 0 2 3 4 9

5 8 1 9 7 6 9 6 7 0 6 3 8 6 8 7 3 8 9 3 7 2 4 4 1

In Exercises 21–28, use the same list of cell phone radiation levels given for Exercises 17–20. Find the indicated percentile or quartile.

P50

Sullivan Survey Choose any quantitative variable from the SullivanStatsSurveyI at www.pearsonhighered.com/sollivanstats. Now choose a qualitative variable, such as gender or political philosophy. Determine the range and standard deviation by the qualitative variable chosen. For example, if you chose gender as the qualitative variable, determine the range and standard deviation by gender. Does there appear to be any difference in the measure of dispersion for each level of the qualitative variable?

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.

California Smokers In the California Health Interview Survey, randomly selected adults are interviewed. One of the questions asks how many cigarettes are smoked per day, and results are listed below for 50 randomly selected respondents. How well do the results reflect the smoking behavior of California adults?

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Roller Coaster z Score A larger sample of 92 roller coaster maximum speeds has a mean of 85.9 km/h and a standard deviation of 28.7 km/h. What is the z score for a speed of 34 km/h? Does the z score suggest that the speed of 34 km/h is significantly low?

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.

Smart Thermostats Listed below are selling prices (dollars) of smart thermostats tested by Consumer Reports magazine. Are any of the resulting statistics helpful in selecting a smart thermostat for purchase?

250 170 225 100 250 250 130 200 150 250 170 200 180 250