Deck 13: Nonparametric Statistics

Question 1

Provide an appropriate response.

-Describe the Wilcoxon rank-sum test. What type of hypotheses is it used to test? What assumptions are made for this test? What is the underlying concept?

Answer 1

The Wilcoxon rank-sum test also looks at ranks but not signs for the data points. The test is used to test claims about the differences between two independent samples. The assumptions include: two independent samples; testing the null hypothesis that the two independent samples come from the same distribution; and more than 10 scores in each of the samples. The underlying principle is that if two samples are drawn from identical populations and the individual scores are all ranked as one combined collection of values, then the high and low ranks should fall evenly between the two samples. For example, if low ranks are found predominantly in one sample with the high ranks in the other, then we suspect that the two samples are not identical.

Question 2

Provide an appropriate response.

-Describe the sign test. What types of hypotheses is it used to test? What is the underlying concept?

Answer 2

The sign test compares the signs (negative or positive)of the differences for data sets, ignoring any ties resulting in a difference of zero. The sign test can be used to test claims involving two dependent samples, claims involving nominal data, and claims about the median of a single population. The underlying concept is that if two sets of data have equal medians, the number of positive signs should be approximately equal to the number of negative signs.

Question 3

Provide an appropriate response.

-List the advantages and disadvantages of nonparametric tests.

Answer 3

Advantages: 1)Nonparametric methods can be applied to a wide variety of situations because they do not have the rigid requirements of their parametric counterparts. In particular, nonparametric tests do not require normally distributed populations. 2)Nonparametric tests can often be applied to nonnumerical data. 3)Nonparametric methods usually involve simpler computations than the corresponding parametric methods.
Disadvantages: 1)Nonparametric methods tend to waste information because exact numerical data are reduced to a qualitative form. 2)Nonparametric tests are not as efficient as parametric tests so we generally need stronger
evidence (such as a larger sample or a greater difference)before we reject a null hypothesis.

Question 4

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?

Question 5

Fourteen people rated two brands of soda on a scale of 1 to 5. $$\begin{array}{l}\begin{array} { l l | l l l l l l l } \text { Brand } & \text { A } & 2 & 3 & 2 & 4 & 3 & 1 & 2 \\\hline \text { Brand } & \text { B } & 1 & 4 & 5 & 5 & 1 & 2 & 3\end{array}\\\\\begin{array} { l l | l l l l l l l } \text { Brand } & \text { A } & 5 & 4 & 2 & 1 & 1 & 4 & 3 \\\hline \text { Brand } & \text { B } & 4 & 5 & 5 & 2 & 4 & 5 & 4\end{array}\end{array}$$ At the 5 percent level, test the null hypothesis that the two brands of soda are equally popular.

Question 6

A researcher wishes to test whether a particular diet has an effect on blood pressure. The blood pressure of 25 randomly selected adults is measured. After one month on the diet, each person's blood pressure is again measured. For 16 people, the second blood pressure reading was lower than the first, and for 9 people, the second blood pressure reading was higher than the first. At the 0.01 significance level, test the claim that the diet has an effect on blood pressure.

Question 7

The heights of 16 randomly selected women are given below. Use a significance level of 0.05 to test the claim that the population median is equal to 64.0 inches. $$\begin{array} { l l l l l l l l } 62.9 & 61.9 & 66.4 & 68.5 & 63.7 & 64.0 & 65.2 & 67.0 \\70.2 & 65.3 & 64.0 & 60.3 & 64.3 & 66.9 & 65.0 & 63.8\end{array}$$

Question 8

The systolic blood pressure readings of ten subjects before and after following a particular diet for a month are shown in the table. Use a significance level of 0.01 to test the claim that the diet has no effect on systolic blood pressure. $$\begin{array} { l l l l l l l l l l l l } \text { Subject } & & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } & \text { G } & \text { H } & \text { I } & \text { J } \\\hline \text { Before } & & 175 & 192 & 167 & 180 & 161 & 203 & 185 & 176 & 204 & 146 \\\hline \text { After } & & 160 & 190 & 170 & 180 & 153 & 197 & 191 & 174 & 192&150\end{array}$$

Question 9

Provide an appropriate response.

-Describe the Kruskal-Wallis test. What types of hypotheses is it used to test? What assumptions are made for this test?

Question 10

Provide an appropriate response.

-Describe the runs test for randomness. What types of hypotheses is it used to test? Does the runs test measure frequency? What is the underlying concept?

Question 11

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. $$\begin{array} { l l l l l l l l l l l l l l l l } \text { Before } & 54 & 61 & 56 & 41 & 38 & 57 & 42 & 71 & 88 & 42 & 36 & 23 & 22 & 46 & 51 \\\hline \text { After } & 82 & 87 & 84 & 76 & 79 & 87 & 42 & 97 & 99 & 74 & 85 & 96 & 69 & 84 & 79\end{array}$$

Question 12

Provide an appropriate response.

-Describe parametric and nonparametric tests. Explain why nonparametric tests are important.

Question 13

A researcher wishes to study whether music has any effect on the ability to memorize information. 87 randomly selected adults are given a memory test in a quiet room. They are then given a second memory test while listening to classical music. 62 people received a higher score on the second test, 24 a lower score, and 1 received the same score. At the 0.05 significance level, test the claim that the music has no effect on memorization skills.

Question 14

A standard aptitude test is given to several randomly selected programmers, and the scores are given below for the mathematics and verbal portions of the test. Use the sign test to test the claim that programmers do better on the mathematics portion of the test. Use a 0.05 level of significance. $$\begin{array} { l l l l l l l l l l l } \hline \text { Mathematics } &347& 440 & 327 & 456 & 427 & 349 & 377 & 398 & 425 \\\hline \text { Verbal } & 285 & 378 & 243 & 371 & 340 & 271 & 294 & 322 & 385\end{array}$$

Question 15

A researcher wishes to study whether a particular diet is effective in helping people to lose weight. 86 randomly selected adults were weighed before starting the diet and again after following the diet for one month. 47 people lost weight, 37 gained weight, and 2 observed no change in their weight. At the 0.01 significance level, test the claim that the diet is effective.

Question 16

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. $$\begin{array} { l l l l l l l l l l } \text { Mathematics} &347 & 440 & 327 & 456 & 427 & 349 & 377 & 398 & 425 \\\hline \text { Verbal } & 285 & 378 & 243 & 371 & 340 & 271 & 294 & 322 & 385\end{array}$$

Question 17

Provide an appropriate response.

-Define rank. Explain how to find the rank for data which repeats (for example, the data set: 4, 5, 5, 5, 7, 8, 12, 12, 15, 18).

Question 18

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 Wilcoxon's signed-ranks test. $$\begin{array} { l l l l l l l l l l l l l l l l } \text { Before } & 54 & 61 & 56 & 41 & 38 & 57 & 42 & 71 & 88 & 42 & 36 & 23 & 22 & 46 & 51 \\\hline \text { After } & 82 & 87 & 84 & 76 & 79 & 87 & 42 & 97 & 99 & 74 & 85 & 96 & 67 & 84 & 79\end{array}$$

Question 19

The waiting times (in minutes)of 28 randomly selected customers in a bank are given below. Use a significance level of 0.05 to test the claim that the population median is equal to 5.3 minutes. $$\begin{array} { r r r r r r r } 8.2 & 8.0 & 10.5 & 3.8 & 6.4 & 5.3 & 7.8 \\2.9 & 6.0 & 7.7 & 6.1 & 5.9 & 1.2 & 10.4 \\7.3 & 6.9 & 5.8 & 5.1 & 6.2 & 3.1 & 5.8 \\11.7 & 4.5 & 6.5 & 9.8 & 7.4 & 2.3 & 7.8\end{array}$$

Question 20

Provide an appropriate response.

-Describe the rank correlation test. What types of hypotheses is it used to test? How does the rank correlation coefficient rs differ from the correlation coefficient r?

Question 21

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. $\begin{array}{llllllllllll}\text { College A } & 3.2 & 4.0 & 2.4 & 2.6 & 2.0 & 1.8 & 1.3 & 0.0 & 0.5 & 1.4 & 2.9 \\\hline \text { College B } & 2.4 & 1.9 & 0.3 & 0.8 & 2.8 & 3.0 & 3.1 & 3.1 & 3.1 & 3.5 & 3.5\end{array}$

Question 22

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. $$\begin{array} { c c c | c c c } &\text { Female } &&&{ \text { Male } } \\\hline 350 & 420 & 470 & 4710 & 460 & 650 \\385 & 675 & 52 & 545 & 720 & 810 \\540 & 400 & 550 & 660 & 500 & 880 \\& 450 & 640 & & 700 & 750\end{array}$$

Question 23

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?

Question 24

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. $$\begin{array} { c c c | c c c } & { \text { Route 1 } } &&& { \text { Route } } 2 \\\hline 35 & 42 & 4 & 41 & 46 & 38 \\33 & 48 & 40 & 49 & 53 & 36 \\39 & 50 & 46 & 51 & 57 & 53 \\& 36 & 40 & 45 & 50 & 55\end{array}$$

Question 25

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. $$\begin{array} { l l l l l l l } \text { Brand X16.4 } & 17.6 & 18.3 & 17.0 & 17.1 & 17.3 & \\\hline \text { Brand Y17.9 } & 18.0 & 17.8 & 18.4 & 17.6 & 19.0 & 19.1 \\\hline \text { Brand Z17.3 } & 16.4 & 16.5 & 16.0 & 15.8 & 16.3 & 17.1\end{array}$$

Question 26

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.

Question 27

The table below shows the lifetimes (in hours)of random samples of light bulbs of three different brands. Use a 0.01 significance level to test the claim that the samples come from identical populations.
$$\begin{array} { r | r | r } \text { Brand A } & \text { Brand B } & \text { Brand C } \\\hline 190 & 182 & 203 \\220 & 170 & 210 \\230 & 203 & 199 \\215 & 175 & 200 \\224 & 178 & 196 \\231 & 181 & 197\end{array}$$

Question 28

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. $\begin{array}{ll}\text { Before } & 12.1&12 .4&11 .7&11 .5&11 .0&11 .8&12 .3&10 .8&12 .6&12 .7&10 .7 \\\hline \text { After } & 11.9&12 .4&11 .8&11 .4&11 .2&11 .5&12 .0&10 .9&12 .01&2 .2&11 .1\end{array}$

Question 29

In a study of the effectiveness of physical exercise in weight reduction, 12 subjects followed a program of physical exercise for two months. Their weights (in pounds)before and after this program are shown in the table. Use a significance level of 0.05 to test the claim that the exercise program has no effect on weight. $$\begin{array} { l l } \text { Before } & 162&190&188&152&148&127&195&164&175&156&180&136 \\\hline \text { After } & 157&194&179&149&135&130&183&168&168&148&170&138\end{array}$$

Question 30

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. $$\begin{array} { l l l l l l } 140 & 116 & 125 & 120 & 153 & 140 \\111 & 127 & 133 & 137 & 132 & 160\end{array}$$

Question 31

Listed below are grade averages for randomly selected students with three different categories of high-school background. At the 0.05 level of significance, test the claim that the three groups come from identical populations.
$\text { HIGH SCHOOL RECORD }$
$\begin{array}{ccc}\text { Good } & \text { Fair } & \text { Poor } \\\hline 3.21 & 2.87 & 2.01 \\3.65 & 3.05 & 2.31 \\1.00 & 2.00 & 2.98 \\3.12 & 0.00 & 0.50 \\2.75 & 1.98 & 2.36\end{array}$

Question 32

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

Question 33

In the sign test procedure the most common approach to handling ties is to exclude the ties. A second approach is to treat half the 0s (representing ties)as positive signs and half as negative signs. In this approach, if the number of ties is odd, one tie is excluded so that they can be divided equally. In a sign test for matched pairs with a claim that the median of the differences is equal to zero, there are 34 positive signs, 54 negative signs, and 23 ties. Identify the test statistic and conclusion for the two different methods. Use a significance level of 0.05.

Question 34

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. $$\begin{array} { c c c | c c c } &\text { school A } &&&{ \text { school B } } \\\hline 550&480&670&460&580&620\\400&700&520&880&680&570\\540&740&560&660&500&480\\360&560&650&&600&550\end{array}$$

Question 35

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. $$\begin{array} { rrrr | rrrr } &&\text { Program I } &&& \text { Program II } \\\hline 60&75&61&63 & 66&89&68&77 \\86&69&64&70 & 84&80&81&87 \\&72&82&59 & 78&73&91&93 \\&&&&&& 94&95\end{array}$$

Question 36

The table below shows the weights (in pounds)of 6 randomly selected women in each of three different age groups. Use a 0.01 significance level to test the claim that the 3 age-group populations of weights are identical. $$\begin{array} { r | r | r } 18 - 34 & 35 - 55 & 56 \text { and older } \\\hline 119 & 123 & 140 \\134 & 147 & 128 \\114 & 135 & 59 \\125 & 110 & 134 \\153 & 154 & 120 \\138 & 163 & 116\end{array}$$

Question 37

Given that the rank correlation coefficient, rs, for 37 pairs of data is 0.373, test the claim of no correlation between the two variables. Use a significance level of 0.01.

Question 38

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. $$\begin{array} { rcr | rcr | rcr } &\text { School A }& && \text { School B } &&& \text { School C }\\\hline 550&480&670&500&620&700&460&580&620\\400&60&520&&550&760&380&6020&470\\&&&&&&&&450\end{array}$$

Question 39

Given that the rank correlation coefficient, rs, for 75 pairs of data is -0.783, test the claim of no correlation between the two variables. Use a significance level of 0.05.

Question 40

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

Question 41

Ten luxury cars were ranked according to their comfort levels and their prices. $$\begin{array} { c c c } \hline \text { Make } & \text { Comfort } & \text { Price } \\\hline \mathbf { A } & 5 & 1 \\\text { B } & 8 & 7 \\\text { C } & 9 & 3 \\\text { D } & 1 . & 5 \\\mathbf { E } & 4 & 4 \\\mathbf { F } & 3 & 2 \\\text { G } & 2 & 16 \\\text { H } & 1 & 9 \\\text { I } & 7 & 6 \\\mathbf { J } & 6 & 8 \\\hline\end{array}$$ Find the rank correlation coefficient and test the claim of no correlation between comfort and price. Use a significance level of 0.05.

Question 42

Given that the rank correlation coefficient, rs, for 20 pairs of data is 0.809, test the claim of no correlation between the two variables. Use a significance level of 0.05.

Question 43

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. $$\begin{array} { c c c c c c c c c c } 94 & 96 & 97 & 99 & 95 & 90 & 97 & 98 & 100 & 100 \\92 & 95 & 98 & 99 & 102 & 97 & 97 & 101 & 99 & 100 \\98 & 95 & 93 & 99 & 101 & 99 & 101 & 100 & 99 & 103\end{array}$$

Question 44

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.

Question 45

A pollster interviews voters and claims that her selection process is random. Listed below is the sequence of voters identified according to gender. At the 0.05 level of significance, test her claim that the sequence is random according to the criterion of gender. $$\begin{array} { l } \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { F } , \mathrm { F } , \mathrm { F } , \mathrm { F } \\\mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { M } , \mathrm { F } , \mathrm { F } , \mathrm { F } , \mathrm { F } , \mathrm { F } , \mathrm { F }\end{array}$$

Question 46

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 $\alpha = 0.05$ .

Question 47

Suppose that a Kruskal-Wallis Test is to be performed and that there are three samples each of size six. What is the largest possible value of the test statistic H?

A)14.75
B)16.89
C)15.16
D)16.25

Question 48

Ten trucks were ranked according to their comfort levels and their prices. $$\begin{array} { c c c } \hline \text { Make } & \text { Comfort } & \text { Price } \\\hline \text { A } & 1 & 6 \\\text { B } & 6 & 2 \\\text { C } & 2 & 3 \\\text { D } & 8 & 1 \\\mathbf { E } & 4 & 4 \\\text { F } & 7 & 8 \\\text { G } & 9 & 10 \\\text { H } & 10 & 9 \\\text { I } & 3 & 5 \\\mathbf { J } & \mathbf { 5 } & 7 \\\hline\end{array}$$ Find the rank correlation coefficient and test the claim of no correlation between comfort and price. Use a significance level of 0.05.

Question 49

In using the Kruskal-Wallis test, there is a correction factor that should be applied whenever there are many ties: Divide H by $$1 - \frac { \sum \mathrm { T } } { \mathrm { N } ^ { 3 } - \mathrm { N } }$$ For each group of tied observations, calculate $\mathrm { T } = \mathrm { t } ^ { 3 } - \mathrm { t }$ , where $\mathrm { t }$ is the number of observations that are tied withi the individual group. Find $\mathrm { t }$ for each group of tied values, then compute the value of $\mathrm { T }$ for each group, then add the T values to get $\sum \mathrm { T }$ . The total number of observations in all samples combined is N. Find the corrected value of $\mathrm { H }$ for the data below which represents test scores for three different groups.
$\begin{array}{lllllll}\text { Group 1: } & 20 & 18 & 18 & 18 & 20 & 20 \\\text { Group 2: } & 18 & 12 & 18 & 18 & 20 & \\\text { Group 3: } & 16 & 18 & 13 & 18 & 18 &\end{array}$

A) $3.69$
B) $- 0.19$
C) $4.91$
D) $3.78$

Question 50

Answers to a questionnaire were in the following sequence. Test for randomness.
$\begin{array} { l l l l l l l l l l } \mathrm { Y } & \mathrm { Y } & \mathrm { N } & \mathrm { Y } & \mathrm { N } & \mathrm { N } & \mathrm { N } & \mathrm { N } & \mathrm { Y } & \mathrm { Y } \\N & N & N & N & Y & Y & Y & N & N & N \end{array}$

Question 51

A placement test is required for students desiring to take a finite mathematics course at a university. The instructor of the course studies the relationship between students' placement test score and final course score. A random sample of eight students yields the following data. $$\begin{array}{l}\text { Placement Score}&\text { Final Course Score }\\\hline 38 & 63 \\90 & 41 \\95 &54 \\51 & 32 \\86 & 93 \\74 & 60 \\60 & 61 \\57 & 89\end{array}$$ Compute the rank correlation coefficient, rs, of the data and test the claim of no correlation between placement score and final course score. Use a significance level of 0.05.

Question 52

Test the sequence of digits below for randomness above and below the value of 4.5. $$\begin{array} { l l l l l l l l l l } 0 & 4 & 7 & 3 & 6 & 0 & 9 & 7 & 4 & 8 \\7 & 2 & 8 & 5 & 7 & 3 & 9 & 6 & 4 & 6 \\4 & 7 & 9 & 1 & 6 & 1 & 9 & 5 & 8 & 3 \\7 & 8 & 5 & 7 & 3 & 5 & 2 & 9 & 3 & 8\end{array}$$

Question 53

A sample of 30 clock radios is selected in sequence from an assembly line. Each radio is examined and judged to be acceptable (A)or defective (D). The results are shown below. Test for randomness. $\begin{array}{cccccccccc}\mathrm{A} & \mathrm{A} & \mathrm{D} & \mathrm{A} & \mathrm{A} & \mathrm{A} & \mathrm{D} & \mathrm{A} & \mathrm{A} & \mathrm{D} \\\mathrm{A} & \mathrm{D} & \mathrm{A} & \mathrm{A} & \mathrm{A} & \mathrm{D} & \mathrm{A} & \mathrm{A} & \mathrm{A} & \mathrm{A} \\\mathrm{A} & \mathrm{A} & \mathrm{D} & \mathrm{D} & \mathrm{A} & \mathrm{A} & \mathrm{A} & \mathrm{A} & \mathrm{D} & \mathrm{A}\end{array}$

Question 54

Test the sequence of digits below for randomness of odd and even digits. $$\begin{array} { l l l l l l l l l l } 0 & 4 & 7 & 3 & 6 & 0 & 9 & 7 & 4 & 8 \\7 & 2 & 8 & 5 & 7 & 3 & 9 & 6 & 4 & 6 \\4 & 7 & 9 & 1 & 6 & 1 & 9 & 5 & 8 & 3 \\7 & 8 & 5 & 7 & 3 & 5 & 2 & 9 & 3 & 8\end{array}$$

Question 55

A sample of 15 clock radios is selected in sequence from an assembly line. Each radio is examined and judged to be acceptable (A)or defective (D). The results are shown below. Test for randomness.
D D A A A
A A A A A
A A D D D

Question 56

Use the sample data below to find the rank correlation coefficient and test the claim of no correlation between math and verbal scores. Use a significance level of 0.05. $$\begin{array} { l l l l l l l l l l l } \text { Mathematics }&347 & 440 & 327 & 456 & 427 & 349 & 377 & 398 & 425 \\\hline \text { Verbal } & 285 & 378 & 243 & 371 & 340 & 271 & 294 & 322 & 385\end{array}$$

Question 57

Use a 0.05 level of significance to test the claim that the sequence of computer-generated numbers is random. Test for randomness above and below the mean. $$8\quad7\quad5\quad7\quad3\quad9\quad1\quad8\quad0\quad4\quad3\quad8\quad4\quad6\quad2\quad3\quad9\quad7\quad5$$

Question 58

A true-false test had the following answer sequence.
T T T T F T F T F T F T T F T
T T F F F F F F F T F T F T F
Test the null hypothesis that the sequence was random.

Question 59

The outcomes (odd number or even number)of a roulette wheel are shown below. Test for randomness of odd (O)and even (E)numbers. $$\begin{array} { l l l l l l l l l l l l } O & \text { E } & O & E & O & \text { E } & O & E & \text { E } & \text { O } & \text { E } & \text { E } \\E & \text { E } & O & \text { E } & O & O & E & O & E & E & O & E\end{array}$$

Question 60

Given that the rank correlation coefficient, rs, for 15 pairs of data is -0.623, test the claim of no correlation between the two variables. Use a significance level of 0.01.

Question 61

When performing a rank correlation test, one alternative to using Table A-9 to find critical values is to compute them using this approximation: $$r _ { S } = \pm \sqrt { \frac { t ^ { 2 } } { t ^ { 2 } + n - 2 } }$$
where $t$ is the $t - s c o r e$ from Table $\mathrm { A } - 3$ corresponding to $n - 2$ degrees of freedom. Use this approximation to find critical values of $r _ { \mathrm { S } }$ for the case where $\mathrm { n } = 11$ and $\alpha = 0.01$ .

A) $\pm 0.685$
B) $\pm 0.411$
C) $\pm 0.726$
D) $\pm 0.735$

Question 62

When performing a rank correlation test, one alternative to using Table A-9 to find critical values is to compute them using this approximation: $$r _ { S } = \pm \sqrt { \frac { t ^ { 2 } } { t ^ { 2 } + n - 2 } }$$
where $t$ is the t-score from Table A-3 corresponding to $\mathrm { n } - 2$ degrees of freedom. Use this approximation to find critical values of $\mathrm { r } _ { \mathrm { S } }$ for the case where $\mathrm { n } = 7$ and $\alpha = 0.05$ .

A) $\pm 0.569$
B) $\pm 0.755$
C) $\pm 0.448$
D) $\pm 0.669$

Question 63

When performing a rank correlation test, one alternative to using Table A-9 to find critical values is to compute them using this approximation: $$r _ { S } = \pm \sqrt { \frac { t ^ { 2 } } { t ^ { 2 } + n - 2 } }$$
where $t$ is the $t$ -score from Table A-3 corresponding to $\mathrm { n } - 2$ degrees of freedom. Use this approximation to find critical values of $\mathrm { r } _ { \mathrm { S } }$ for the case where $\mathrm { n } = 17$ and $\alpha = 0.05$ .

A) $\pm 0.411$
B) $\pm 0.480$
C) $\pm 0.311$
D) $\pm 0.482$

Question 64

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

A)A
B)C
C)D
D)B