MATH 220 Assignment #7 complete solutions correct answers key

MATH 220 Assignment #7 complete solutions correct answers key

Question 1: (MINITAB)

Twelve runners are asked to run a 10-kilometer race on each of two consecutive weeks.  In one of the races, the runners wear one brand of shoe and in the other a different brand.  The brand of shoe they wear in which race is determined at random.  All runners are timed and are asked to run their best in each race.  The results (in minutes) are given below.

Runner

Brand 1

Brand 2

1

31.23

32.02

2

29.33

28.98

3

30.50

30.63

4

32.20

32.67

5

33.08

32.95

6

31.52

31.53

7

30.68

30.83

8

31.05

31.10

9

33.00

33.12

10

29.67

29.50

11

30.55

30.57

12

32.12

32.20

Note that in this case the data are paired (the same subject is used to generate a pair of observations.) The data are analyzed by focusing on the sample of paired differences (paired diff. = obs. before ‘minus’ obs. after). This essentially reduces the problem to a one-sample t-procedure under the right assumptions.

a)      Make a scatterplot of the running times of brand 1 vs. brand 2. Was it a good idea to pair data? Explain.

b)      Use a statistical test to test the hypothesis that that athletes tend to run faster in a 10-kilometer race using brand 1 than using brand 2.

c)      Construct a 95% confidence interval for the difference between the mean time to run a 10-kilometer race using brand 1 and the mean time to run a 10-kilometer race using brand 2.

d)     Use a boxplot of the paired differences to assess the validity of the t procedure you used to analyze the data.

Minitab Instructions:

a)      Graph → Plot → etc.

The following set of commands will give you the output you need to answer all the questions b), c),and d):

-          Choose Stat → Basic Statistics → Paired t

-          Enter the Brand 1 data in the 'First sample' box

-          Enter the Brand 2 data in the 'Second sample' box

-          Choose 'Options'

-          Enter 95 in the 'Confidence level' box and press Tab

-          Enter 0 in the 'Test mean' box (the no difference hypothesis) and press Tab

-          Note that the default alternative (the following box) is ‘Not equal’. Leave it as is and press Enter

-          Choose ‘Graphs’

-          Select ‘Boxplot of differences’ and press Enter twice

The output will consist of

-          The boxplot for the paired differences (needed for the graphical check).

-          The results of the testing procedure (to be found in the session window) for a non-directional test. You may have to halve the P-value if your test is directional.

-          The confidence interval.

Copy the output to your MS Word file and answer the questions.

Question 2: (MINITAB)

Read the background information on the Wade Tract in Thomas County, Georgia, on page 352 (8e) [page 342 (7e)] (Example 6.1). The tract is divided into eastern and western halves. Thirty longleaf pine trees are randomly selected from each half. The diameter at breast height (DBH) is measured for each sample (data provided separately).

a) Obtain side-by-side histograms, boxplots, and summary statistics. 

b) Examine the data graphically. Describe any patterns you see.

b) Is there evidence that the mean DBH of the eastern half differs from the mean DBH of the western half.  State the null and alternative hypotheses clearly. Report the value of the test statistic and the P value of the test (you can insert the appropriate part of the Minitab output). State your conclusions in terms relevant to the setting.

c) Find a 95% confidence interval for the mean difference in mean DBH’s. What additional information is the confidence interval providing (relative to the test  you performed in b)?

d) Are the t procedures used to answer in b) and c) appropriate?  Explain your answer.

Minitab Instructions:

We are comparing two independent samples.

a) To obtain the graphical and numerical displays:

-          Choose 'Stat → Basic Statistics → Display Descriptive Statistics'

-          Select the DBH variable for the ‘Variables’ box

-          Select the ‘By variable’ box and enter the ‘Area’ variable

-          Click the ‘Graphs’ dialog box open

-          Select ‘Histogram of data’

-          Select ‘Boxplot of data’ and press Enter twice.

b) and c) To perform a two-sample t test:

-          Choose Stat → Basic Statistics → 2-Sample t

-          Enter the DBH variable in the 'Samples' box

-          Enter the ‘Area’ variable in the 'Subscripts' box

-          Choose 'Options'

-          Enter the confidence level in the 'Confidence level' box

-           Enter 0 in the 'Test difference' box (the no difference hypothesis)

-          Leave the ‘not equal’ alternative as is in the 'Alternative' menu.

-          Press Enter twice.

The output will consist of

-          The results of the testing procedure (to be found in the session window) for a non-directional test. You may have to halve the P-value if your test is directional.

-          The confidence interval.

d)  Address the assumptions for (and the appropriateness of) the t procedure.

Question 3: (EXCEL/MINITAB)

The data classifies a sample of 865 college students according to their field of study and their score on a scale called PEOPLE designed to measure altruism defined as an interest in the welfare of others. For this scale, lower scores were designed to indicate lower levels of altruism. The table below summarizes the results.

School

LOW

MEDIUM

HIGH

Agriculture

5

27

35

Consumer/Family Sciences

1

32

54

Engineering

12

129

94

Liberal Arts

7

77

129

Management

3

44

28

Science

7

29

24

Technology

2

62

64

a)      Find the conditional distributions of the variable ALTRUISM for each school.

b)      Construct the bar graphs of the three conditional distributions on the same page (Excel does this very nicely).

c)      Is level of altruism independent of field of study? Perform a chi-square test for independence. Be sure to state your hypotheses and conclusions.

Minitab Instructions:

a) Transform each count into a percent relative to row total. (Excel?)

b)  Reproduce the table (with percents in place of counts) in Excel and obtain the side-by-side bar graphs from Excel.

c) Switch now to Minitab:

Recreate the original table of counts in C1 (LOW), C2 (MEDIUM), and C3 (HIGH).

To find the value of  the Chisquare statistic:

-          Choose Stat → Tables → Chisquare Test

-          Type C1-C3 in the Columns containing table box and press Enter

The output gives you the chisquare value as well as the P-value of the test. Make sure you provide a complete answer by writing null and alternative hypotheses, value of the chi-square statistic, and P value.