The scores of twelve students on the midterm exam and the final exam were as follows. \[\begin{array} { l c c } \hline \text { Student } & \text { Midterm } & \text { Final } \\ \hline \text { Navarro } & 93 & 91 \\ \text { Reaves } & 89 & 85 \\ \text { Hurlburt } & 71 & 73 \\ \text { Knuth } & 65 & 77 \\ \text { Lengyel } & 62 & 67 \\ \text { Mcmeekan } & 74 & 79 \\ \text { Bolker } & 77 & 65 \\ \text { Ammatto } & 87 & 83 \\ \text { Pothakos } & 82 & 89 \\ \text { Sul1 ivan } & 81 & 71 \\ \text { Wahl } & 91 & 81 \\ \text { Zurfluh } & 83 & 94 \\ \hline \end{array}\] Find the rank correlation coefficient and test the claim of no correlation between midterm score and final exam score. Use a significance level of 0.05.

@#LAT-DLM&. Critical values: @#LAT-DLM& Significan

A college administrator collected information on first-semester night-school students. A random sample taken of 12 students yielded the following data on age and GPA during the first semester. \[\begin{array}{l} \begin{array} { c c c } \text { Age}&& \text {GPA }\\ { x } & & y \\ \hline 18 & & 1.2 \\ 26 & & 3.8 \\ 27 & & 2.0 \\ 37 & & 3.3 \\ 33 & & 2.5 \\ 47 & & 1.6 \\ 20 & & 1.4 \\ 48 & & 3.6 \\ 50 & & 3.7 \\ 38 & & 3.4 \\ 34 & & 2.7 \\ 22 & & 2.8 \end{array} \end{array}\] Do the data provide sufficient evidence to conclude that the variables age, $\mathrm { x }$, and GPA, $\mathrm { y }$, are correlated? Apply ; rank-correlation test. Use $\alpha = 0.05$.

@#LAT-DLM& No significant correlation. T

SAT scores for students selected randomly from three different schools are shown below. Use a significance level of 0.05 to test the claim that the samples come from identical populations. \[\begin{array} { rcr | rcr | rcr } &\text { School A }& && \text { School B } &&& \text { School C }\\ \hline 550&480&670&500&620&700&460&580&620\\400&60&520&&550&760&380&6020&470\\&&&&&&&&450 \end{array}\]

Test statistic: H = 3.6586. Critical val

Exam 13: Nonparametric Statistics

Use the Wilcoxon signed-ranks test and the sample data below. At the 0.05 significance level, test the claim that math and verbal scores are the same. Mathematics 347 440 327 456 427 349 377 398 425 Verbal 285 378 243 371 340 271 294 322 385

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Question 1

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Test statistic T = 0. Critical value: T = 6.
Reject the null hypothesis that both samples come from the same population distribution.

A teacher uses two different CAI programs to remediate students. Results for each group on a standardized test are listed in a table below. At the 0.05 level of significance, test the hypothesis that the two programs produce different results. Program I Program II 60 75 61 63 66 89 68 77 86 69 64 70 84 80 81 87 72 82 59 78 73 91 93 94 95

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Question 2

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$\mu _ { R } = 143 , \sigma _ { R } = 18.2665 , R = 90 , z = - 2.90 .$
Test statistic: $z = - 2.90$ . Critical values $z = \pm 1.96$ .
Reject the null hypothesis. There is sufficient evidence to support the hypothesis that the two programs pr different results.

The Mann-Whitney U test is equivalent to the Wilcoxon rank-sum test for independent samples in the sense that they both apply to the same situations and always lead to the same conclusions. In the Mann-Whitney U test we calculate $z = \frac { U - \frac { \mathrm { n } _ { 1 } \mathrm { n } _ { 2 } } { 2 } } { \sqrt { \frac { \mathrm { n } _ { 1 } \mathrm { n } _ { 2 } \left( \mathrm { n } _ { 1 } + \mathrm { n } _ { 2 } + 1 \right) } { 12 } } }$ where $\mathrm { U } = \mathrm { n } _ { 1 } \mathrm { n } _ { 2 } + \frac { \mathrm { n } _ { 1 } \left( \mathrm { n } _ { 1 } + 1 \right) } { 2 } - \mathrm { R }$ For the sample data below, use the Mann-Whitney U test to test the null hypothesis that the two independent samples come from populations with the same distribution. State the hypotheses, the value of the test statistic, the critical values, and your conclusion. Test scores (men): 70, 96, 77, 90, 81, 45, 55, 68, 74, 99, 88 Test scores (women): 89, 92, 60, 78, 84, 96, 51, 67, 85, 94

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Question 3

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H?: The two samples come from populations with the same distribution.
H?: The two samples come from populations with different distributions. Critical values $\mathrm { z } = \pm 1.96$
Test statistic: $\mathrm { z } = 0.39$
Do not reject the null hypothesis. There is not sufficient evidence to reject the claim that the two samples come
from populations with the same distribution.

A rank correlation coefficient is to be calculated for a collection of paired data. The values lie between -10 and 10. Which of the following could affect the value of the rank correlation coefficient? A: Multiplying every value of one variable by 3 B: Interchanging the two variables C: Adding 2 to each value of one variable D: Replacing every value of one variable by its absolute value

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Question 4

How does the Wilcoxon rank-sum test compare to the corresponding t-test in terms of efficiency, ease of calculations and assumptions required? Are there any kinds of data for which the Wilcoxon rank-sum test can be used but the t-test cannot be used?

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Question 5

Construct a data set with n = 14 such that the sign test would lead to rejection of the null hypothesis that the median is equal to 50 while the t-test conclusion is failure to reject the null hypothesis of µ = 50.

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Question 6

The sequence of numbers below represents the maximum temperature (in degrees Fahrenheit)in July in one U.S. town for 30 consecutive years. Test the sequence for randomness above and below the median. 94 96 97 99 95 90 97 98 100 100 92 95 98 99 102 97 97 101 99 100 98 95 93 99 101 99 101 100 99 103

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Question 7

11 runners are timed at the 100-meter dash and are timed again one month later after following a new training program. The times (in seconds)are shown in the table. Use a significance level of 0.05 to test the claim that the training has no effect on the times. Before 12.1 12.4 11.7 11.5 11.0 11.8 12.3 10.8 12.6 12.7 10.7 After 11.9 12.4 11.8 11.4 11.2 11.5 12.0 10.9 12.01 2.2 11.1

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Question 8

The scores of twelve students on the midterm exam and the final exam were as follows. Student Midterm Final Navarro 93 91 Reaves 89 85 Hurlburt 71 73 Knuth 65 77 Lengyel 62 67 Mcmeekan 74 79 Bolker 77 65 Ammatto 87 83 Pothakos 82 89 Sul1 ivan 81 71 Wahl 91 81 Zurfluh 83 94 Find the rank correlation coefficient and test the claim of no correlation between midterm score and final exam score. Use a significance level of 0.05.

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Question 9

11 female employees and 11 male employees are randomly selected from one company and their weekly salaries are recorded. The salaries (in dollars)are shown below. Use a significance level of 0.10 to test the claim that salaries for female and male employees of the company have the same distribution. Female Male 350 420 470 4710 460 650 385 675 52 545 720 810 540 400 550 660 500 880 450 640 700 750

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Question 10

Ten trucks were ranked according to their comfort levels and their prices. Make Comfort Price A 1 6 B 6 2 C 2 3 D 8 1 4 4 F 7 8 G 9 10 H 10 9 I 3 5 7 Find the rank correlation coefficient and test the claim of no correlation between comfort and price. Use a significance level of 0.05.

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Question 11

A college administrator collected information on first-semester night-school students. A random sample taken of 12 students yielded the following data on age and GPA during the first semester. Age GPA x y 18 1.2 26 3.8 27 2.0 37 3.3 33 2.5 47 1.6 20 1.4 48 3.6 50 3.7 38 3.4 34 2.7 22 2.8 Do the data provide sufficient evidence to conclude that the variables age, $\mathrm { x }$ , and GPA, $\mathrm { y }$ , are correlated? Apply ; rank-correlation test. Use $\alpha = 0.05$ .

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Question 12

A fire-science specialist tests three different brands of flares for their burning times (in minutes)and the results are given below for the sample data. At the 0.05 significance level, test the claim that the three brands have the same mean burn time. Use the Kruskal-Wallis test. Brand X16.4 17.6 18.3 17.0 17.1 17.3 Brand Y17.9 18.0 17.8 18.4 17.6 19.0 19.1 Brand Z17.3 16.4 16.5 16.0 15.8 16.3 17.1

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Question 13

An instructor gives a test before and after a lesson and results from randomly selected students are given below. At the 0.05 level of significance, test the claim that the lesson has no effect on the grade. Use the sign test. Before 54 61 56 41 38 57 42 71 88 42 36 23 22 46 51 After 82 87 84 76 79 87 42 97 99 74 85 96 69 84 79

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Question 14

SAT scores for students selected randomly from two different schools are shown below. Use a significance level of 0.05 to test the claim that the scores for the two schools have the same distribution. school A school B 550 480 670 460 580 620 400 700 520 880 680 570 540 740 560 660 500 480 360 560 650 600 550

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Question 15

The Wilcoxon signed-ranks test can be used to test the claim that a sample comes from a population with a specified median. The procedure used is the same as the one described in this section except that the differences are obtained by subtracting the value of the hypothesized median from each value. The sample data below represent the weights (in pounds)of 12 women aged 20-30. Use a Wilcoxon signed-ranks test to test the claim that the median weight of women aged 20-30 is equal to 130 pounds. Use a significance level of 0.05. Be sure to state the hypotheses, the value of the test statistic, the critical values, and your conclusion. 140 116 125 120 153 140 111 127 133 137 132 160

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Question 16

A person who commutes to work is choosing between two different routes. He tries the first route 11 times and the second route 12 times and records the time of each trip. The results (in minutes)are shown below. Use a significance level of 0.01 to test the claim that the times for both routes have the same distribution. Route 1 Route 2 35 42 4 41 46 38 33 48 40 49 53 36 39 50 46 51 57 53 36 40 45 50 55

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Question 17

Provide an appropriate response. -Describe the Wilcoxon signed-ranks test. What types of hypotheses is it used to test? What assumptions are made for this test?

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Question 18

SAT scores for students selected randomly from three different schools are shown below. Use a significance level of 0.05 to test the claim that the samples come from identical populations. School A School B School C 550 480 670 500 620 700 460 580 620 400 60 520 550 760 380 6020 470 450

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Question 19

Use the Wilcoxon rank-sum approach to test the claim that students at two colleges achieve the same distribution of grade averages. The sample data is listed below. Use a 0.05 level of significance. College A 3.2 4.0 2.4 2.6 2.0 1.8 1.3 0.0 0.5 1.4 2.9 College B 2.4 1.9 0.3 0.8 2.8 3.0 3.1 3.1 3.1 3.5 3.5

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Question 20

Use the Wilcoxon signed-ranks test and the sample data below. At the 0.05 significance level, test the claim that math and verbal scores are the same. Mathematics 347 440 327 456 427 349 377 398 425 Verbal 285 378 243 371 340 271 294 322 385

How does the Wilcoxon rank-sum test compare to the corresponding t-test in terms of efficiency, ease of calculations and assumptions required? Are there any kinds of data for which the Wilcoxon rank-sum test can be used but the t-test cannot be used?

Construct a data set with n = 14 such that the sign test would lead to rejection of the null hypothesis that the median is equal to 50 while the t-test conclusion is failure to reject the null hypothesis of µ = 50.

The sequence of numbers below represents the maximum temperature (in degrees Fahrenheit)in July in one U.S. town for 30 consecutive years. Test the sequence for randomness above and below the median. 94 96 97 99 95 90 97 98 100 100 92 95 98 99 102 97 97 101 99 100 98 95 93 99 101 99 101 100 99 103

Ten trucks were ranked according to their comfort levels and their prices. Make Comfort Price A 1 6 B 6 2 C 2 3 D 8 1 4 4 F 7 8 G 9 10 H 10 9 I 3 5 7 Find the rank correlation coefficient and test the claim of no correlation between comfort and price. Use a significance level of 0.05.

Provide an appropriate response. -Describe the Wilcoxon signed-ranks test. What types of hypotheses is it used to test? What assumptions are made for this test?

SAT scores for students selected randomly from three different schools are shown below. Use a significance level of 0.05 to test the claim that the samples come from identical populations. School A School B School C 550 480 670 500 620 700 460 580 620 400 60 520 550 760 380 6020 470 450

Introduction to Statistics

Summarizing and Graphing Data

Statistics for Describing, Exploring, and Comparing Data

Probability

Probability Distributions

Normal Probability Distributions

Estimates and Sample Sizes

Hypothesis Testing

Inferences From Two Samples

Correlation and Regression

Multinomial Experiments and Contingency Tables

Analysis of Variance

Statistical Process Control

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