3. Describing Data Numerically

Degrees of Freedom Five recent U.S. presidents had a mean age of 56.2 years at the time of their inauguration. Four of these ages are 64, 46, 54, and 47.

a. Find the missing value.

Trimmed Mean Because the mean is very sensitive to extreme values, we say that it is not a resistant measure of center. By deleting some low values and high values, the trimmed mean is more resistant. To find the 10% trimmed mean for a data set, first arrange the data in order, then delete the bottom 10% of the values and delete the top 10% of the values, then calculate the mean of the remaining values. Use the axial loads (pounds) of aluminum cans listed below (from Data Set 41 “Aluminum Cans” in Appendix B) for cans that are 0.0111 in. thick. An axial load is the force at which the top of a can collapses. Identify any outliers, then compare the median, mean, 10% trimmed mean, and 20% trimmed mean.

247 260 268 273 276 279 281 283 284 285 286 288

289 291 293 295 296 299 310 504

Body Temperatures Listed below are body temperatures (°F) of adult males (based on Data Set 5 “Body Temperatures” in Appendix B).

97.6 98.2 99.6 98.7 99.4 98.2 98.0 98.6 98.6

a. Find the mean. Does the result seem reasonable?

Exercises 1–10 are based on the following sample data consisting of costs of dinner (dollars) and the amounts of tips (dollars) left by diners. The data were collected by students of the author.

Predictions Repeat the preceding exercise assuming that the linear correlation coefficient is r = 0.132.

In Exercises 5–8, use the following two control charts that result from testing batches of newly manufactured aircraft altimeters, with 100 in each batch. The original sample values are errors (in feet) obtained when the altimeters are tested in a pressure chamber that simulates an altitude of 6000 ft. The Federal Aviation Administration requires an error of no more than 40 ft at that altitude.

What is the value of x_double bar In general, how is a value of xbar found?

Quadratic Mean The quadratic mean (or root mean square, or R.M.S.) is used in physical applications, such as power distribution systems. The quadratic mean of a set of values is obtained by squaring each value, adding those squares, dividing the sum by the number of values n, and then taking the square root of that result, as indicated below:

Quadratic mean = sqrt(∑x^2/n)

Find the R.M.S. of these voltages measured from household current: 0, 60, 110, 0. How does the result compare to the mean?

Cloud Seeding The “Florida Area Cumulus Experiment” was conducted by using silver iodide to seed clouds with the objective of increasing rainfall. For the purposes of this exercise, let the daily amounts of rainfall be represented by units of rnfl. (The actual rainfall amounts are in or )

Find the value of the following statistics and include appropriate units based on rnfl as the unit of measurement.

15.53 7.27 7.45 10.39 4.70 4.50 3.44 5.70 8.24 7.30 4.05 4.46

a. mean

b. median

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).

35 35 20 50 95 75 45 50 30 35 30 30

a. Find the mean xbar.

Birth Weights The table below lists some of the same data used in the preceding exercise, but the seven days of the week are combined into weekday (Monday, Tuesday, Wednesday, Thursday, Friday) and weekend days (Saturday, Sunday). Also, the birth weights are converted to kilograms. What do you conclude?

For the four test scores 96, 85, 91, and 86, the first 3 test scores are 20% of the final grade, and the last test score is 40% of the final grade. Find the weighted mean of the test scores.

According to data from the city of Toronto, Ontario, Canada, there were nearly 112,000 parking infractions in the city for December 2020, with fines totaling over 5,500,000 Canadian dollars. The fines (in Canadian dollars) for a random sample of 105 parking infractions in Toronto, Ontario, Canada, for December 2020 are listed below. (Source: City of Toronto)

In Exercises 1–5, use technology. If possible, print your results.

Find the sample mean of the data.

Finding a Weighted Mean In Exercises 41– 46, find the weighted mean of the data.

Grades A student receives the grades shown below, with an A worth 4 points, a B worth 3 points, a C worth 2 points, and a D worth 1 point. What is the student’s grade point average?

Grades In Exercise 46, one of the student’s B grades gets changed to an A. What is the student’s new grade point average?

Finding the Mean of a Frequency Distribution In Exercises 49–52, approximate the mean of the frequency distribution.

Social Media The average daily amounts of time (in minutes) spent on Snapchat

Finding the Mean of a Frequency Distribution In Exercises 49–52, approximate the mean of the frequency distribution.

Populations The populations (in thousands) of the counties in Montana in 2019 (Source: U.S. Census Bureau)