AP* Statistics: 8th Edition

Many real-life applications of statistics involve comparisons of two populations. Such comparisons can involve modifications of graphical displays such as dotplots, bar charts, stemplots, boxplots, and cumulative frequency plots to portray both sets simultaneously.

When asked for a comparison, don’t forget to address shape, outliers (unusual values), center and spread (SOCS or CUSS), and to refer to context. You must use comparative words, that is, you must state which center and which spread is larger (or if they are approximately the same). Simply making two separate lists is not enough and will be penalized.

The caloric intakes of 25 people on each of two weight loss programs are recorded as follows:

Program A: 1000, 1000, 1100, 1100, 1100, 1200, 1200, 1200, 1200, 1300, 1300, 1300, 1300, 1300, 1400, 1400, 1400, 1400, 1400, 1500, 1500, 1600, 1600, 1700, 1900

Program B: 1000, 1100, 1100, 1200, 1200, 1200, 1300, 1300, 1300, 1400, 1400, 1400, 1400, 1500, 1500, 1500, 1500, 1500, 1600, 1600, 1600, 1700, 1700, 1800, 1800

These data can be compared with dotplots, one above the other, using the same horizontal scale.

Program A appears to be associated with a lower average caloric intake than Program B. Comparing shape, center, and spread, we have:

Shape: We see that both sets of data are roughly bell-shaped (the empirical rule applies), and Program A has an outlier at 1900 calories (while Program B has no outliers).

Center: Visually, or by counting dots, the centers of the two distributions are 1300 and 1400 calories, for Programs A and B respectively. (A calculator gives means of x_A = 1336 and x_B = 1424.) By any method, the center for Program B is higher.

Spread: The spreads are approximately the same, 1000 to 1900 calories for Program A and 1000 to 1800 calories for Program B. (A calculator gives standard deviations of s_A = 218 and s_B = 218.)

A study tabulated the percentages of young adults who recognized various photographs as follows: Joe Stalin (10%), Joe Camel (95%), Senator Simpson of Wyoming (5%), Bart Simpson (80%), Al Gore (30%), Al Bundy (60%), Mickey Mantle (20%), Mickey Mouse (100%), Charlie Chaplin (25%), Charlie the Tuna (90%). These data can be illustrated with a bar chart appropriately displayed in pairs of bars.

The pairs of bar graphs visually indicate the reason for concern felt by some educators.

In a 40-year study, survival years were measured for cancer patients undergoing one of two different chemotherapy treatments. The data for 25 patients on the first drug and 30 on the second were as follows:

Drug A: 5, 10, 17, 39, 29, 25, 20, 4, 8, 31, 21, 3, 12, 11, 19, 10, 4, 22, 17, 18, 13, 28, 11, 14, 21

Drug B: 19, 12, 20, 28, 22, 35, 1, 21, 21, 26, 18, 28, 29, 20, 15, 32, 31, 24, 22, 26, 18, 20, 22, 35, 30, 18, 25, 24, 19, 21

In drawing a back-to-back stemplot of the above data, we place a vertical line on each side of the column of stems and then arrange one set of leaves extending out to the right while the other extends out to the left.

Note that even though drug A showed the longest-surviving patient (39 years) and drug B showed the shortest-surviving patient (1 year), the back-to-back stemplot indicates that the bulk of patients on drug B survived longer than the bulk of patients on drug A.

Shape: Both distributions are roughly bell-shaped (the empirical rule applies). The drug A distribution appears to have a high outlier at 39, while the drug B distribution appears to have a low outlier at 1.

Center: Visually, or by counting values, the centers of the two distributions are 17 and 22, respectively. (A calculator gives means of x_A = 16.48 and x_B = 22.73.) By either method, the drug B distribution has a greater center.

Spread: The spreads are 3 to 39 survival years for drug A and 1 to 35 survival years for drug B. (A calculator gives standard deviations of s_A = 9.25 and s_B = 6.97.) By either method, the drug A distribution has a greater spread than the drug B distribution.

Mail-order labs and 1-hour minilabs were compared with regard to price for developing and printing one 24-exposure roll of 35-millimeter color-print film. Prices included shipping and handling charges where applicable. Following is a computer output describing the results:

Mean = 5.37 Standard deviation = 1.92 Min = 3.51
Max = 8.00 N = 18 Median = 4.77
Quartiles = 3.92, 6.45

Mean = 10.11 Standard deviation = 1.32 Min = 8.58
Max = 11.95 N = 15 Median = 10.08
Quartiles = 8.97, 11.51

In drawing parallel boxplots (also called side-by-side boxplots) of the above data, we place both on the same diagram:

Boxplots show the minimum, maximum, median, and quartile values. The distribution of mail-order lab prices is lower and more spread out than that of prices of 1-hour minilabs. Both are slightly skewed toward the upper end (the skewness can also be noted from the computer output showing the mean to be greater than the median in both cases.)

Parallel boxplots are useful in presenting a picture of the comparison of several distributions.

Following are parallel boxplots showing the daily price fluctuations of a certain common stock over the course of 5 years. What trends do the boxplots show?

The parallel boxplots show that from year to year the median daily stock price has steadily risen 20 points from about $58 to about $78, the third quartile value has been roughly stable at about $84, the yearly low has never decreased from that of the previous year, and the interquartile range has never increased from one year to the next.

The graph below compares cumulative frequency plotted against age for the U.S. population in 1860 and in 1980.

Answer: Looking across from .5 on the vertical axis, we see that in 1860 half the population was under the age of 20, while in 1980 all the way up to age 32 must be included to encompass half the population. Looking across from .25 and .75 on the vertical axis, we see that for 1860, Q₁ = 9 and Q₃ = 35 and so the interquartile range is 35 – 9 = 26 years, while for 1980, Q₁ = 16 and Q₃ = 50 and so the interquartile range is 50 – 16 = 34 years. Thus, both the median and the interquartile range were greater in 1980 than in 1860.

For all the above, make note of any similarities and differences in shape, center, and spread.

Directions: The questions or incomplete statements that follow are each followed by five suggested answers or completions. Choose the response that best answers the question or completes the statement.

1. The dotplots below show the yearly wages of all male and female executives at a large firm.

(A) A greater proportion of male employees than female employees are executives at this firm.

(D) The range of salaries paid to male executives is less than the range of salaries paid to female executives.

3. Consider the following back-to-back stemplots comparing car battery lives (in months) of samples of two popular brands.

4. The following boxplots were constructed from SAT math scores of boys and girls at a high school:

Which of the following is a possible boxplot for the combined scores of all the students?

Questions 5–7 refer to the following population pyramids (source: U.S. Census Bureau).

II. The recent civil war in Liberia, with the extensive use of child soldiers, has had an impact on the population age distribution.

8. Looking at very large sets of communications metadata, mainly phone call and e-mail logs, a government agency tracks connections starting with intelligence targets overseas and extending to “contact chains” of different lengths. During January 2014 and January 2015, the lengths of contact chains analyzed are shown in the following histograms.

In which month did the chain length distribution have the greater standard deviation?

Directions: You must show all work and indicate the methods you use. You will be graded on the correctness of your methods and on the accuracy of your final answers.

1. Are women better than men at multitasking? Suppose in one study of multitasking a random sample of 200 female and 200 male high school students were assigned several tasks at the same time, such as solving simple mathematics problems, reading maps, and answering simple questions while talking on a telephone. Total times taken to complete all the tasks are given in the histograms below.

Write a few sentences comparing the distributions of times to complete all tasks by females and by males.

2. In independent random samples of 20 men and 20 women, the number of minutes spent on grooming on a given day were:

Men: 27, 32, 82, 36, 43, 75, 45, 16, 23, 48, 51, 57, 60, 64, 39, 40, 69, 72, 54, 57

Women: 49, 50, 35, 69, 75, 35, 49, 54, 98, 58, 22, 34, 60, 38, 47, 65, 79, 38, 42, 87

3. To analyze the social media behavior differences between boys and girls, Mrs. V’s FDA high school AP Statistics class was asked to count the number of text messages that they sent over a long three-day weekend. The following table summarizes the data:

	Values under Q₁	Q₁	Median	Q₃	Values over Q₃
Females	15, 43, 100	130	175	358	450, 573, 1098
Males	3, 59	72	183	273	293, 337

(b) Do the data indicate that females or males had the greater mean number of texts? Explain.

4. Cumulative frequency graphs of the ages of people on three different Caribbean cruises (A, B, and C) are given below:

Write a few sentences comparing the distributions of ages of people on the three cruises.

The NFL quarterback rating formula provides a means of comparing passing performances. The graphs below show the quarterback ratings for two players during one 16-game season.

(d) What information is more apparent from the above 16-game graphs than from the dotplots?

A central moving average is calculated using data equally spaced either side of the point in the series where the mean is calculated. The first few lines of the 3-game central moving averages for the quarterback ratings of the two players are as follows:

(e) Fill in the two blank spaces corresponding to the fourth game above. Show your calculations.