SECTION 16.1 Exercises

For Exercises 16.1 and 16.2, see page 16-5; for 16.3 and 16.4, see page 16-6; for 16.5 and 16.6, see page 16-8; for 16.7, see page 16-11; for 16.8, see page 16-12.

16-13

Question 16.9

16.9 Price Discrimination?

CASE 16.1 In Examples 16.1 through 16.4, the testing of significance was based on the rank sum of the Used Auto consumers being the first group. Suppose we associate the first group with the Midwest Auto consumers. As noted in the section (page 16-5), the rank sum for Midwest Auto consumers is 55.

used

  1. Find the mean and standard deviation of the rank sum for Midwest Auto consumers under the null hypothesis that markup percents do not differ significantly.
  2. Refer to Example 16.1 where the mean of the rank sum for Used Auto consumers was found. What is the relationship between the two means?
  3. In Example 16.3, the one-sided -value was determined using the rank sum of Used Auto consumers. Show how the same reported -value can be found using the rank sum of Midwest Auto consumers.

16.9

(a) , . (b) , which is the sum of the ranks. (c) , , which is the same as in Example 16.3.

Question 16.10

16.10 Wheat prices.

Example 7.13 (pages 385-386) reports the results of a small survey that asked separate samples of 5 wheat producers in each of January and July what price they received for wheat sold that month. Here are the data:

wheat

Month Price of wheat ($/bushel)
January $6.6125 $6.4775 $6.3500 $6.7525 $6.7625
July $6.7350 $6.9000 $6.6475 $7.2025 $7.0550

The stemplot on page 386 shows a large difference between months. We cannot assess Normality from such small samples. Carry out by hand the steps in the Wilcoxon rank sum test for comparing prices in January and July.

  1. Arrange the 10 observations in order and assign ranks. There are no ties.
  2. Find the rank sum for July. What are the mean and standard deviation of under the null hypothesis that prices in January and July do not differ systematically?
  3. Standardize to obtain a statistic. Do a Normal probability calculation with the continuity correction to obtain a two-sided -value.

Question 16.11

16.11 Online discussion posting.

Students in a fully online MBA statistics course are required as to post relevant learning contributions in the course’s discussion forum throughout the semester. These posts serve as a form of online participation and factor into their grades. Below find the number of posts during the semester by the six female students in the class.5

posts

23 27 20 16 10 31

Find the ranks for these data.

16.11

There are no tied values; the following table shows the corresponding ranks.

Sex M M M M F M M M
Posts 3 4 5 9 10 12 13 14
Rank 1 2 3 4 5 6 7 8
Sex M F M M F F F F
Posts 15 16 17 18 20 23 27 31
Rank 9 10 11 12 13 14 15 16

Question 16.12

16.12 Find the rank sum statistic.

Refer to the previous exercise. Here are the data for the 10 men in the class.

posts

18 9 14 15 13 17 4 12 5 3

Compute the value of the Wilcoxon statistic. Take the first sample to be the women.

Question 16.13

16.13 State the hypotheses.

Refer to the previous exercise. State appropriate null and alternative hypotheses for this setting.

16.13

H0: There is no difference in distribution of the number of posts between females and males. Ha: One gender has a systematically higher number of posts than the other.

Question 16.14

16.14 Find the mean and standard deviation of the distribution of the statistic.

The statistic that you calculated in Exercise 16.12 is a random variable with a sampling distribution. What are the mean and the standard deviation of this sampling distribution under the null hypothesis?

Question 16.15

16.15 Find the -value.

Refer to Exercises 16.11 through 16.14. Find the -value using the Normal approximation with the continuity correction and interpret the result of the significance test.

16.15

, . There is evidence of a systematic difference in the number of posts by gender.

Question 16.16

16.16 Counts of seeds in one-pound scoops.

Exercise 7.55 (page 395) discusses a study of two different packaging plants in terms of the packaging of seeds. An SRS of 50 one-pound scoops of seeds was collected from Plant 1746, and an SRS of 19 one-pound scoops of seeds was collected from Plant 1748. The number of seeds found in each scoop was recorded. Histograms and Normal quantile plots suggest the data arise from non-Normal distributions. Using software, is there a significant difference in the number of seeds per pound between the two plants based on the Wilcoxon test?

seedcnt2

Question 16.17

16.17 Polyester fabrics in landfills.

How quickly do synthetic fabrics such as polyester decay in landfills? A researcher buried polyester strips in the soil for different lengths of time, then dug up the strips and measured the force required to break them. Breaking strength is easy to measure and is a good indicator of decay. Lower strength means the fabric has decayed. Part of the study involved burying 10 polyester strips in well-drained soil in the summer. Five of the strips, chosen at random, were dug up after two weeks; the other five were dug up after 16 weeks. Here are the breaking strengths in pounds:6

poly

2 weeks 118 126 126 120 129
16 weeks 124 98 110 140 110
  1. Make side-by-side stemplots for the two groups. Does it appear reasonable to assume that the two distributions have the same shape?
  2. Is there evidence that breaking strengths are lower for strips buried longer?

16.17

(a) The distributions are very different and not the same shape. (b) From software, The data do not show a systematic difference in breaking strengths.

16-14

Question 16.18

16.18 Economic growth.

The most commonly used measure of economic growth is the rate of growth in a country’s total output of goods and services gauged by the gross domestic product (GDP) adjusted for inflation. The level of a country’s GDP growth reflects on the growth of businesses, jobs, and personal income. Here are World Bank data on the average growth of GDP (percent per year) for the period 2010 to 2013 in developing countries of Europe:7

reggdp

Country Growth Country Growth
Albania 2.3 Macedonia, FYR 2.1
Armenia 4.4 Moldova 5.5
Azerbaijan 3.2 Montenegro 1.7
Belarus 4.0 Romania 1.3
Bosnia and Herzegovina 0.4 Serbia 0.9
Bulgaria 0.9 Turkey 6.0
Georgia 5.6 Ukraine 2.9
Kosovo 3.4

Here are the data for nearby developing countries of the Central Asia:

Country Growth Country Growth
Uzbekistan 8.2 Kyrgyz Republic 4.0
Turkmenistan 11.3 Kazakhstan 6.5
Tajikistan 7.2
  1. Arrange the 20 observations in order and assign ranks. Be aware of ties.
  2. What is the rank sum for the Central Asia region?
  3. Perform the two-sided test at the level, making sure to report the test statistic, and -value. What is your conclusion?

Question 16.19

16.19 It’s your choice.

Exercise 16.18 asks for the rank sum for Central Asia.

reggdp

  1. What is the rank sum for European developing countries? The ranks of 20 observations always add to 210. Do your two sums add to 210?
  2. Repeat the previous exercise using the the rank sum for European countries. Show that you obtain exactly the same -value. That is, your choice between the two possible ’s does not affect the results of the Wilcoxon test.

16.19

(a) . Yes, . (b) , .

Question 16.20

16.20 ERP implementation.

Companies worldwide are investing in enterprise resource planning (ERP) systems. ERP is an integrated business management software system that allows companies to share common data across all functional business areas. By linking areas of a company with a single system, the premise is that companies will reduce costs and improve efficiencies corporate-wide. In a study, researchers investigated if ERP implementation had positive impact on facility management (FM) services. A survey was conducted on companies with and without ERP systems. Data were collected by company on the number of FM-related areas that have had productivity improvements in the given calendar year. The researchers were interested in testing the alternative hypothesis that the number of FM-related productivity improvements was greater for ERP companies than non-ERP companies.8

erp

  1. Examine the data from each group. Explain why a two-sample test may not be the best choice for conducting a test between the two groups.
  2. There are many ties among the observations. Arrange the observations in order and assign ranks, assigning all tied values the average of the ranks they occupy.
  3. What is the rank sum for the ERP group?
  4. Perform the appropriate one-sided test at the level, making sure to report the test statistic and -value. What is your conclusion?

Question 16.21

16.21 The influence of subliminal messages.

Can “subliminal” messages that are below our threshold of awareness nonetheless influence us? Advertisers, among others, want to know. One study asked if subliminal messages help students learn math. A group of students who had failed the mathematics part of the City University of New York Skills Assessment Test agreed to participate in a study to find out. All received a daily subliminal message, flashed on a screen too rapidly to be consciously read. The treatment group of 10 students was exposed to “Each day I am getting better in math.” The control group of eight students was exposed to a neutral message, “People are walking on the street.” All students participated in a summer program designed to raise their math skills, and all took the assessment test again at the end of the program. Here are data on the subjects’ scores before and after the program.9

sublim

Treatment group Control group
Pretest Posttest Pretest Posttest
18 24 18 29
18 25 24 29
21 33 20 24
18 29 18 26
18 33 24 38
20 36 22 27
23 34 15 22
23 36 19 31
21 34
17 27
  1. The study design was a randomized comparative experiment. Outline this design.
  2. Compare the gain in scores in the two groups, using a graph and numerical descriptions. Does it appear that the treatment group’s scores rose more than the scores for the control group?
  3. Apply the Wilcoxon rank sum test to the posttest versus pretest differences. Note that there are some ties. What do you conclude?

16.21

(b) The histograms show that the differences for the Treatment group appear higher than those for the control group. (c) , . The treatment group has systematically higher differences than the control group; the subliminal messages appear to work.

16-15

Question 16.22

16.22 Fitness and ego.

Exercise 7.63 describes a study of fitness and personality. In particular, 28 middle-aged college faculty were evenly divided into low-fitness and high-fitness groups. The subjects then took the Cattell Sixteen Personality Factor Questionnaire. The provided data are the measurements of “ego strength.”

ego

  1. Arrange the observations in order and assign ranks. Assign any tied values the average of the ranks they occupy.
  2. What is the rank sum for the high-fitness group?
  3. Perform the significance test at the level, making sure to report the test statistic, and -value. What is your conclusion?

Question 16.23

16.23 Safety of restaurant food.

CASE 16.2 Case 16.2 (page 16-10) describes a study of the attitudes of people attending outdoor fairs about the safety of the food served at such locations. In the associated data file, you will find the responses of 303 people to several questions. The variables in this data set are (in order)

subject hfair sfair sfast srest gender

The variable “sfair” contains the responses described in the example concerning safety of food served at outdoor fairs and festivals. The variable “srest” contains responses to the same question asked about food served in restaurants. We saw that women are more concerned than men about the safety of food served at fairs. Is this also true for restaurants?

fsafety

16.23

, , . Women are systematically more concerned than men about the safety of food served at restaurants.

Question 16.24

16.24 Food safety at fairs and in restaurants.

CASE 16.2 The data file used in Example 16.5 (page 16-11) and Exercise 16.23 contains 303 rows, one for each of the 303 respondents. Each row contains the responses of one person to several questions. We wonder if people are more concerned about the safety of food served at fairs than they are about the safety of food served at restaurants. Explain carefully why we cannot answer this question by applying the Wilcoxon rank sum test to the variables “sfair” and “srest.”

Question 16.25

16.25 Sadness and spending.

Exercise 7.52 (page 394) studies the effect of sadness on a person’s spending judgment. In the exercise, we find 17 participants in the sad group and 14 participants in the neutral group.

sadness

  1. Arrange the 31 observations in order and assign ranks. Be aware of ties.
  2. What is the rank sum for the sad group?
  3. Using software, is there evidence of a sadness effect in that people in a sad mood systematically spend more than people not in a sad mood?

16.25

(b) (c) , People in a sad mood systematically spend more than people not in a sad mood.

Question 16.26

16.26 Patient satisfaction.

A Wisconsin health care provider has a network of hospitals serving communities throughout eastern Wisconsin. In an attempt to continually improve its services, this provider conducts patient and employee satisfaction surveys. To measure overall rating of a given hospital, patients are asked, “Would you recommend this hospital to your friends and family?” Answer choices are definitely no, probably no, probably yes, definitely yes. The responses are coded numerically from 1 to 4. Here are quarterly survey data for one of its urban-based hospitals and for one of its suburban-based hospitals.

hsurvey

Response code Response Urban Suburb
1 definitely no 12 6
2 probably no 33 9
3 probably yes 47 33
4 definitely yes 58 52
  1. Is there a relationship between location and rating? Use the chi-square test to answer this question.
  2. The chi-square test ignores the ordering of the rating categories. The provided data file contains data on the 250 patients surveyed. The first variable is the location (Urban or Suburb) and the second is the rating code as it appears in the table (1 to 4). Is there good evidence that patients in one location have systematically higher satisfaction ratings than in the other?