Question 1

If a student is interested in determining whether or not a particular score was the median
grade on the statistics exam, he might ask several other students what their scores were
then apply the Wilcoxon rank sum test..

Accepted Answer

The answer of If a student is interested in determining...

Question 2

It is believed that children can throw a ball further after lunch than before lunch. A resear، decided to investigate this hypothesis using the paired-sample sign test. Each of 14 childı a ball as far as they could before and after lunch. The distances in meters are tabulated be
$\begin{array}{l|lllllll}
\text { Subject } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \
\hline \text { Before Lunch } & 23 & 29 & 28 & 28 & 29 & 40 & 19 \
\text { After Lunch } & 26 & 35 & 34 & 25 & 35 & 37 & 22
\end{array}$

$\begin{array}{l|lllllll}
\text { Subject } & \mathbf{8} & \mathbf{9} & \mathbf{1 0} & \mathbf{1 1} & \mathbf{1 2} & \mathbf{1 3} & \mathbf{1 4} \
\hline \text { Before Lunch } & 23 & 28 & 56 & 19 & 40 & 28 & 28 \
\text { After Lunch } & 26 & 34 & 65 & 22 & 37 & 25 & 34
\end{array}$

$\text { Test the researcher's hypothesis at } \alpha=0.05 \text {. }$

You may use the following abbreviated table of critical values for the sign test (one tailed $ \alpha=0.05 $ ), taken from Table $ \mathrm{J} $ in the text:

$\begin{array}{l|ccccccccccccccc}
\mathbf{n} & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 \
\hline \text { critical value } & 2 & 2 & 3 & 3 & 3 & 4 & 4 & 5 & 5 & 5 & 6 & 6 & 7 & 7 & 7
\end{array}$
 her
A) The data does not support the claim (does not reject the null hypothesis) because the test value 3 is equal to the critical value 3.
B) The data supports the claim (rejects null hypothesis) because the test value 3 is more than the critical value 2.
C) The data does not support the claim (does not reject the null hypothesis) because the test value 4 is more than the critical value 3.
D) The data does not support the claim (does not reject the null hypothesis) because the test value 4 is less than the critical value 7.

Accepted Answer

The answer of It is believed that children can throw...

Question 3

A random sample of 12 men and 12 women were asked to report the number of hours they watched television on a Wednesday evening. If a researcher wants to test the hypothesis that there is no difference in the amount of time spent watching television based on gender, what $z$-values should be used for a two-tailed Wilcoxon rank sum test, at $\alpha = 0.05$ ?
A) $z = \pm 2.00$
B) $z = \pm 1.67$
C) $z = \pm 3.00$
D) $z = \pm 1.96$

Accepted Answer

For a two-tailed test at $\alpha = 0.05$, the critical $z$-values are $\pm 1.96$, which correspond to the standard normal distribution's critical values for the 2.5th and 97.5th percentiles.

Question 4

The following data was collected as part of a study examining whether there is a difference between the number of hours men and women watch television. The values represent the number of hours a subject watched television on a designated Monday night. In the process of computing the test value the data from both samples should be combined, arranged in order, and ranked according to each group. Calculate the sum of th for both groups. Lower values rank ahead of higher ones.
$\begin{array}{l|llllllllllll}
\text { Men } & 2.0 & 1.5 & 3.0 & 2.5 & 2.0 & 1.0 & 0.0 & 2.0 & 1.5 & 2.5 & 2.0 & 2.0 \
\hline \text { Women } & 2.0 & 2.5 & 1.0 & 1.0 & 1.5 & 2.5 & 2.0 & 1.0 & 2.0 & 1.5 & 1.0 & 0.0
\end{array}$

A) The sum of the ranks for the men is 130, and the sum of the ranks for the woman is 170.
B) The sum of the ranks for the men is 137.5, and the sum of the ranks for the woman is 162.5.
C) The sum of the ranks for the men is 170, and the sum of the ranks for the woman is 130.
D) The sum of the ranks for the men is 162.5, and the sum of the ranks for the woman is 137.5.

Accepted Answer

The answer of The following data was collected as part...

Question 5

A landscaping company hypothesizes that the median number of lawns they mow in a we 20. They tabulated the number of lawns mowed in a random sample of 14 weekends.
$$\begin{array} { l l l l l l l } 
22 & 18 & 19 & 20 & 25 & 20 & 18 \
22 & 23 & 20 & 21 & 26 & 24 & 22
\end{array}$$
What critical value should be used for this two-tailed test at $\alpha = 0.10$ ? d is
A) 2
B) 1
C) 4
D) 3

Accepted Answer

The critical value for a two-tailed test at $\alpha = 0.10$ with 14 observations is 2.

Question 6

A landscaping company hypothesizes that the median number of lawns they mow weekend is 20 . They tabulated the number of lawns mowed in a random sample of weekends.
$\begin{array} { l l l l l l l } 22 & 18 & 19 & 20 & 25 & 20 & 18 \ 22 & 23 & 20 & 21 & 26 & 24 & 22 \end{array}$
State the null hypothesis.

Accepted Answer

The answer of A landscaping company hypothesizes that the median...

Question 7

A researcher needs to conduct a suitable nonparametric test to compare two dependent variables. Which of the following should he use? 
A) Wilcoxon rank sum test
B) Wilcoxon signed-rank test
C) $t$-test
D) Kruskal-Wallis test

Accepted Answer

The Wilcoxon signed-rank test is the correct choice for comparing two dependent variables using a nonparametric approach. This test is used when the data does not meet the assumptions necessary for a parametric test like the $t$-test, specifically when the data are paired or matched in some way (e.g., before-and-after measurements on the same subjects).

Question 8

For a specific year, the median price of natural gas was $\$ 11.99$ per 1000 cubic feet. A researcher wishes to see if there is enough evidence to reject the claim. Out of 43 households, 34 paid less than $\$ 11.99$ per 1000 cubic feet for natural gas. Test the claim at $\alpha = 0.05$.

Accepted Answer

The answer of For a specific year, the median price...

Question 9

The Wilcoxon rank sum test is used to test dependent samples and the Wilcoxon
signed-rank test is used for independent samples.

Accepted Answer

The answer of The Wilcoxon rank sum test is used...

Question 10

Six second grade students tried tossing a ball into a basket ten times each. Their teacher tl suggested a different way of tossing the ball, and the six students tried again. The numbe successful tosses, before and after the teacher's suggestion, are shown below.
$\begin{array}{c|cccccc}
\text { Student } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } \
\hline \text { Before the suggestion (out of 10) } & 5 & 3 & 5 & 4 & 5 & 3 \
\text { After the suggestion (out of 10) } & 2 & 3 & 2 & 6 & 2 & 3
\end{array}$

Find the positive rank sum. [Recall, differences of zero are not ranked.]
A) +3
B) +9
C) +15
D) +1

Accepted Answer

The positive rank sum is calculated by ranking the differences in scores (after - before) and summing the ranks for the positive differences. Here, only Student D improved (6 - 4 = +2), so it gets the highest rank, which is 1. Thus, the positive rank sum is +1. However, the correct calculation involves summing all positive ranks, which in this case, due to the misunderstanding in explanation, should still focus on the fact that only improvements (positive changes) are considered. Given the data, it seems there was a misinterpretation in the initial explanation. The correct approach would involve identifying all improvements and summing their ranks, but based on the provided data, only one student showed improvement, leading to a confusion in the explanation provided.

Question 11

Six second-graders tried tossing a ball into a basket ten times each. Their teacher then sug a different way of tossing the ball, and the six students tried again. The number of succes tosses, before and after the teacher's suggestion, are shown below.
$\begin{array}{c|cccccc}
\text { Student } & \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D} & \mathbf{E} & \mathbf{F} \
\hline \text { Before the suggestion (out of 10) } & 2 & 3 & 7 & 2 & 7 & 3 \
\text { After the suggestion (out of 10) } & 3 & 3 & 5 & 3 & 5 & 3
\end{array}$

Find the sum of the positive and negative rank sums as in a Wilcoxon signed-rank test. gested
A) positive rank sum = 7; negative rank sum = -3
B) positive rank sum = 6; negative rank sum = -2
C) positive rank sum = 4; negative rank sum = -2
D) positive rank sum = 2; negative rank sum = -1

Accepted Answer

The Wilcoxon signed-rank test involves ranking the differences between paired observations (in this case, the number of successful tosses before and after the suggestion), without considering the sign for the purpose of ranking, then summing the ranks for the positive differences and separately for the negative differences. Here, the differences are: +1 (A), 0 (B), -2 (C), +1 (D), -2 (E), 0 (F). Ignoring the signs and zeros, the ranks are 1.5 (A and D) and 3.5 (C and E). The positive rank sum is 1.5 + 1.5 = 3, and the negative rank sum is -3.5 - 3.5 = -7. However, the correct interpretation of the question's request for "the sum of the positive and negative rank sums" would actually be 3 + (-7) = -4, which does not match any of the provided options, indicating a misunderstanding in the calculation or interpretation of the question's request. Given the options and the typical process for calculating Wilcoxon signed-rank test values, none directly align with the correct calculation process described. The correct calculation should consider all differences, assign ranks, and then sum these ranks for positive and negative differences separately. The misunderstanding seems to be in the interpretation of the question's final request rather than the calculation of the rank sums themselves.

Question 12

Two independent samples of tall and short students are selected, and their scores on a 25 point statistics quiz are recorded as shown in the table below. What is the rank of Wilt's s in most exams, higher values rank ahead of lower values.
$\begin{array}{c|cccccccccc}
\text { Tall } & 19 & 21 & 16 & 18 & 14 & 17 \text { (Wilt) } & 12 & 21 & 14 & 16 \
\hline \text { Short } & 14 & 17 & 21 & 19 & 16 & 21 & 18 & 17 & 19 & 18
\end{array}$

A) 13
B) 8
C) 12
D) 7

Accepted Answer

To determine Wilt's rank, we first need to order all the scores from both groups together from highest to lowest. The scores, when combined and sorted, are: 21, 21, 21, 21, 19, 19, 19, 18, 18, 18, 18, 17, 17, 17 (Wilt), 16, 16, 14, 14, 12. Wilt's score of 17 ranks 13th in this ordered list, but since there are three instances of 17, and we rank higher values ahead of lower values, Wilt's rank is shared and considered as the first occurrence of the score 17, making his rank 12th.

Question 13

The following data was collected as part of a study examining whether there is a differenc between the number of hours men and women watch television. The values represent the of hours a subject watched television on a designated Monday night. Lower values rank $\varepsilon$ higher ones.
$\begin{array}{l|llllllllllll}
\text { Men } & 2.0 & 1.5 & 3.0 & 2.5 & 2.0 & 1.0 & 0.0 & 2.0 & 1.5 & 2.5 & 2.0 & 2.0 \
\hline \text { Women } & 2.0 & 2.5 & 1.0 & 1.0 & 1.5 & 2.5 & 2.0 & 1.0 & 2.0 & 1.5 & 1.0 & 0.0
\end{array}$

Calculate $\mu _ { R } , \sigma _ { R }$ and the $z$ test value for the Wilcoxon rank sum test.
A) $\mu _ { R } = 150 , \sigma _ { R } = 5 , z = 4$
B) $\mu _ { R } = 150 , \sigma _ { R } = 17.32 , z = 1.15$
C) $\mu _ { R } = 12.25 , \sigma _ { R } = 12.25 , z = 9.61$
D) $\mu _ { R } = 12.25 , \sigma _ { R } = 150 , z = 0.79$

Accepted Answer

The Wilcoxon rank sum test involves ranking all the observations together, calculating the sum of ranks for one of the groups (usually the smaller one), and then using these ranks to find the mean rank ($\mu_R$) and standard deviation of ranks ($\sigma_R$). The $z$ test value is then calculated to determine if there is a significant difference between the two groups. The correct values for $\mu_R$, $\sigma_R$, and $z$ are given in option B, as it uses the formula for the Wilcoxon rank sum test correctly, considering the size of the sample and the distribution of ranks among the groups.

Question 14

Two independent sets of data are collected, the first with 15 subjects and the second with 26 subjects. Mr. X is a subject in the first data set and is associated with a value of 66.
In the first set, 3 values are ranked behind 66 and 11 values are ranked ahead of it (with
Mr) X being the other data value). In the second set, 9 values are ranked behind 66 and
16 values are ranked ahead of it. What is Mr. X's rank in a Wilcoxon rank sum test?
A) 28
B) 27
C) 12
D) 11

Accepted Answer

In the Wilcoxon rank sum test, ranks are assigned to all values from both data sets combined. Mr. X's value of 66 is ranked 12th in the first set and 10th in the second set. Therefore, Mr. X's combined rank is 12 + 16 = 28.

Question 15

In the Wilcoxon rank sum test, if the hypothesis is true, the
values in the different groups will not be ranked similarly.

Accepted Answer

The answer of In the Wilcoxon rank sum test, if...

Question 16

A researcher believes that the median lowest temperature for February is 25 degrees Fahrenheit. A random sample of 14 years yields the following lowest temperatures for Fel in degrees Fahrenheit:
$\begin{array} { l l l l l l l l l l l l l l } 10 & 10 & 12 & 14 & 14 & 16 & 17 & 19 & 20 & 22 & 25 & 27 & 31 & 33 \end{array}$
Test the researcher's hypothesis at $\alpha = 0.05$.
You may use the following abbreviated table of critical values (two tailed, $\alpha = 0.05$ ) from Table J:
$\begin{array}{l|ccccccccccccccc}
\mathbf{n} & \mathbf{1 1} & \mathbf{1 2} & \mathbf{1 3} & \mathbf{1 4} & \mathbf{1 5} & \mathbf{1 6} & \mathbf{1 7} & \mathbf{1 8} & \mathbf{1 9} & \mathbf{2 0} & \mathbf{2 1} & \mathbf{2 2} & \mathbf{2 3} & \mathbf{2 4} & \mathbf{2 5} \
\hline \text { critical value } & 1 & 2 & 2 & 3 & 3 & 3 & 4 & 4 & 4 & 5 & 5 & 5 & 6 & 6 & 6
\end{array}$

A) Do not reject the claim because the test value 3 is equal to the critical value
B) Reject the claim because the test value 4 is more than the critical value 3.
C) Do not reject the claim because the test value 3 is more than the critical value
D) Reject the claim because the test value 4 is more than the critical value 2.

Accepted Answer

The answer of A researcher believes that the median lowest...

Question 17

Six second grade students tried tossing a ball into a basket ten times each. Their teacher tl suggested a different way of tossing the ball, and the six students tried again. The numbel successful tosses, before and after the teacher's suggestion, are shown below.
$\begin{array}{c|cccccc}
\text { Student } & \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D} & \mathbf{E} & \mathbf{F} \
\hline \text { Before the suggestion (out of 10) } & 5 & 3 & 5 & 2 & 5 & 3 \
\text { After the suggestion (out of 10) } & 3 & 3 & 3 & 3 & 3 & 3
\end{array}$

Find the negative rank sum. [Recall, differences of zero are not ranked.]
A) -3.5
B) -1
C) -9
D) -1.5

Accepted Answer

The answer of Six second grade students tried tossing a...

Question 18

A landscaping company hypothesizes that the median number of lawns they mow in a we 20 . They tabulated the number of lawns mowed in a random sample of 14 weekends.
$\begin{array} { l l l l l l l } 22 & 18 & 19 & 20 & 25 & 20 & 18 \ 22 & 23 & 20 & 21 & 26 & 24 & 22 \end{array}$
Use the test value, the critical value, and the appropriate table to decide if the landscaping company's hypothesis that they mow a median of 20 lawns per weekend valid at $\alpha$. d is
A) Yes. The test value is less than the critical value, so the null hypothesis is not rejected
B) No. The test value is greater than the critical value, so the null hypothesis is rejected.
C) Yes. The test value is greater than the critical value, so the null hypothesis is not rejected.
D) No. The test value is less than the critical value, so the null hypothesis is rejected

Accepted Answer

The answer of A landscaping company hypothesizes that the median...

Question 19

A landscaping company hypothesizes that the median number of lawns they mow in a we 20. They tabulated the number of lawns mowed in a random sample of 14 weekends.
$\begin{array} { l l l l l l l } 22 & 18 & 19 & 20 & 25 & 20 & 18 \ 22 & 23 & 20 & 21 & 26 & 24 & 22 \end{array}$
What is the test value? d is
A) 7
B) 8
C) 4
D) 3

Accepted Answer

To find the test value (d) for a hypothesis test concerning the median, we first need to rank the data and then count how many data points are above and below the hypothesized median. The hypothesized median is 20. Counting the numbers less than 20, we have 5 (18, 18, 19, 19, 20), and counting the numbers greater than 20, we have 9. The test value (d) is the smaller of these two counts, which is 3.

Question 20

It was thought that hypnosis would lower a person's desire to smoke. A researcher decided to investigate this hypothesis using a paired-sample sign test in which the number of cigarettes/day smoked by subjects was recorded before and after hypnosis. Given that the critical value at $\alpha = 0.05$ in a one-tailed test is 1 , is there enough evidence support the hypothesis?
$\begin{array}{c|cccccccccc}
\text { Subject } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \
\hline \text { Before Hypnosis (Cigarettes/day) } & 18 & 20 & 25 & 15 & 19 & 21 & 30 & 16 & 13 & 20 \
\text { After Hypnosis (Cigarettes/day) } & 16 & 18 & 24 & 10 & 20 & 23 & 26 & 15 & 14 & 18
\end{array}$

A) Since the critical value is greater than the test value, there is enough evidence to reject the null hypothesis. Consequently, the data support the hypothesis that
Hypnosis reduces smoking.
B) Since the critical value is greater than the test value, there is not enough evidence to reject the null hypothesis. Consequently, the data do not support the hypothesis that
Hypnosis reduces smoking.
C) Since the critical value is less than the test value, there is enough evidence to reject the null hypothesis. Consequently, the data support the hypothesis that hypnosis
Reduces smoking.
D) Since the critical value is less than the test value, there is not enough evidence to reject the null hypothesis. Consequently, the data do not support the hypothesis that
Hypnosis reduces smoking.

Accepted Answer

The answer of It was thought that hypnosis would lower...

Quiz 13: Nonparametric Statistics

If a student is interested in determining whether or not a particular score was the median grade on the statistics exam, he might ask several other students what their scores were then apply the Wilcoxon rank sum test..

A researcher needs to conduct a suitable nonparametric test to compare two dependent variables. Which of the following should he use?

For a specific year, the median price of natural gas was $\$ 11.99$ per 1000 cubic feet. A researcher wishes to see if there is enough evidence to reject the claim. Out of 43 households, 34 paid less than $\$ 11.99$ per 1000 cubic feet for natural gas. Test the claim at $\alpha = 0.05$ .

The Wilcoxon rank sum test is used to test dependent samples and the Wilcoxon signed-rank test is used for independent samples.

In the Wilcoxon rank sum test, if the hypothesis is true, the values in the different groups will not be ranked similarly.