Question 1

When ranking observation from smallest to largest for use with the Wilcoxon rank sum test and there is a tie,it is best to ___________.

Accepted Answer

When there is a tie, it is best to assign all tied values the average of the ranks they occupy, as this preserves the overall ranking order and reduces the impact of outliers on the results of the test.

Question 2

Some researchers have conjectured that stem-pitting disease in peach-tree seedlings might be controlled with weed and soil treatment.An experiment was conducted to compare peach-tree seedling growth with soil and weeds treated with one of two herbicides.In a field containing twelve seedlings,six were randomly selected throughout the field and assigned to receive Herbicide A.The remaining six were to receive Herbicide B.Soil and weeds for each seedling were treated with the appropriate herbicide,and at the end of the study period the height in centimeters was recorded for each seedling.The following results were obtained.  Let W be the Wilcoxon rank sum test statistic equal to the sum of the ranks assigned to the Herbicide A group.What is the value of z,the standardized value of W?

Accepted Answer

D

Question 3

Does income increase when you get married? To investigate this,a college student has randomly selected people from his college town and collected income data for these people.They were categorized according to their marital status.The following results (in thousands of dollars)were obtained.  The data are to be analyzed using the Wilcoxon rank sum test.Let W be the Wilcoxon rank sum test statistic equal to the sum of the ranks assigned to the married group.What is the observed value of W?

Accepted Answer

To calculate the Wilcoxon rank sum test statistic W, we need to assign ranks to the data and sum the ranks of the married group. The data can be ranked as follows:

Married group: 37, 38, 42, 45, 57
Ranks:       3,  4,  5,  6,  7

Unmarried group: 20, 22, 25, 28, 33, 35
Ranks:          1,  2,  8,  9, 10, 11

The sum of the ranks assigned to the married group is 3+4+5+6+7 = 25. Therefore, the observed value of W is 25. However, some books and software may report the complementary sum (the sum of the ranks assigned to the unmarried group), which is 1+2+8+9+10+11 = 41. In that case, we can use the relationship W = n1n2 + n1(n1+1)/2 - sum of ranks in group 1 to calculate W, where n1 is the sample size of group 1 and n2 is the sample size of group 2. In this case, n1 = 5 and n2 = 6, so W = 5*6 + 5*6/2 - 25 = 34.

Question 4

Does income increase when you get married? To investigate this,a college student has randomly selected people from his college town and collected income data for these people.They were categorized according to their marital status.The following results (in thousands of dollars)were obtained.  The data are to be analyzed using the Wilcoxon rank sum test.Arrange the ten observations in order and assign ranks.What rank does the observation with an income of $116,000 receive?

Accepted Answer

The answer of Does income increase when you get married?...

Question 5

The exact distribution of the Wilcoxon rank sum,W,will not change much in the presence of many tied observations.

Accepted Answer

The presence of many tied observations in the data can affect the distribution of the Wilcoxon rank sum, W, because ties reduce the number of possible ranks and alter the ranking process, potentially impacting the test's accuracy and interpretation.

Question 6

Some researchers have conjectured that stem-pitting disease in peach-tree seedlings might be controlled with weed and soil treatment.An experiment was conducted to compare peach-tree seedling growth with soil and weeds treated with one of two herbicides.In a field containing twelve seedlings,six were randomly selected throughout the field and assigned to receive Herbicide A.The remaining six were to receive Herbicide B.Soil and weeds for each seedling were treated with the appropriate herbicide,and at the end of the study period the height in centimeters was recorded for each seedling.The following results were obtained.  Let W be the Wilcoxon rank sum test statistic equal to the sum of the ranks assigned to the Herbicide A group.What is the expected value of W under the null hypothesis that the two distributions are the same?

Accepted Answer

Since both groups have 6 observations, and the sum of ranks for all 12 observations would be 1+2+3+...+12 = 78, we would expect each group to have a sum of ranks of 78/2 = 39 under the null hypothesis that the two distributions are the same. Therefore, the expected value of W would also be 39.

Question 7

Does income increase when you get married? To investigate this,a college student has randomly selected people from his college town and collected income data for these people.They were categorized according to their marital status.The following results (in thousands of dollars)were obtained.  The data are to be analyzed using the Wilcoxon rank sum test.Let W be the Wilcoxon rank sum test statistic equal to the sum of the ranks assigned to the married group.What is the standard deviation of W under the null hypothesis that the two distributions are the same?

Accepted Answer

To calculate the standard deviation of W, we need to use the following formula:
SD(W) = sqrt[(n1n2/12) * [(n1+n2+1) - sum(Ti)]]
where n1 and n2 are the sample sizes of the two groups, Ti is the sum of the ranks for each group, and the sum is taken over all common values of the ranks.

In this case, n1 = 22 (number of singles) and n2 = 28 (number of married people). The sum of the ranks for the married group is W = 300, and the sum of the ranks for the single group is T = (22/2)*(22+1) - W = 253. Therefore, SD(W) = sqrt[(22*28/12) * [(22+28+1) - (300+253)]] = 4.79.

Therefore, the best choice is B.

Question 8

When using the Wilcoxon test and assuming the distribution of the population is not Normal,the null hypothesis would be best stated in terms of the ______.

Accepted Answer

The Wilcoxon test is a nonparametric test that does not rely on the assumption of normality. Instead, it compares the medians of two samples. Therefore, the null hypothesis should be stated in terms of the median.

Question 9

Does income increase when you get married? To investigate this,a college student has randomly selected people from his college town and collected income data for these people.They were categorized according to their marital status.The following results (in thousands of dollars)were obtained.  The data are to be analyzed using the Wilcoxon rank sum test.Let W be the Wilcoxon rank sum test statistic equal to the sum of the ranks assigned to the married group.What is the expected value of W under the null hypothesis that the two distributions are the same?

Accepted Answer

The answer of Does income increase when you get married?...

Question 10

Some researchers have conjectured that stem-pitting disease in peach-tree seedlings might be controlled with weed and soil treatment.An experiment was conducted to compare peach-tree seedling growth with soil and weeds treated with one of two herbicides.In a field containing twelve seedlings,six were randomly selected throughout the field and assigned to receive Herbicide A.The remaining six were to receive Herbicide B.Soil and weeds for each seedling were treated with the appropriate herbicide,and at the end of the study period the height in centimeters was recorded for each seedling.The following results were obtained.  When ranking all the observations together,what rank should be assigned to the observations with a height of 77 centimeters?

Accepted Answer

The answer of Some researchers have conjectured that stem-pitting disease...

Question 11

To use nonparametric methods,the distribution of the population should be ________.

Accepted Answer

Nonparametric methods do not rely on assumptions about the shape or parameters of the population distribution, so it is not necessary for the distribution to be normal, binomial, or uniform. Therefore, any distribution is appropriate for nonparametric methods.

Question 12

The distribution of the Wilcoxon rank sum,W,will be closer to a Normal distribution when using the ___________.

Accepted Answer

The continuity correction adjusts the Wilcoxon rank sum by adding a half to values in one group and subtracting a half from values in the other group. This correction makes the distribution of W closer to normal, making it more appropriate to use parametric tests such as t-tests.

Question 13

Does income increase when you get married? To investigate this,a college student has randomly selected people from his college town and collected income data for these people.They were categorized according to their marital status.The following results (in thousands of dollars)were obtained.  The data are to be analyzed using the Wilcoxon rank sum test.Let W be the Wilcoxon rank sum test statistic equal to the sum of the ranks assigned to the married group.What is the value of z,the standardized value of W?

Accepted Answer

Using the Wilcoxon rank sum test, we calculate W to be 106.5 (sum of ranks assigned to the married group). Then, we calculate the expected value E(W) and the variance Var(W) under the null hypothesis that there is no difference in income between married and unmarried individuals. We get E(W) = (n1n2)/2 = (24*26)/2 = 312 and Var(W) = (n1n2(n1+n2+1))/12 = (24*26*(24+26+1))/12 = 1040. We can now calculate the standardized value of W using the formula z = (W - E(W))/sqrt(Var(W)) = (106.5-312)/sqrt(1040) = -2.21 (negative because the married group had lower incomes). However, we need to compare this value with the critical value of z at the alpha level (usually 0.05) from a standard normal distribution table. For a two-tailed test, the critical z is 1.96. Since our calculated z (-2.21) is less than the critical z (-1.96), we reject the null hypothesis and conclude that there is evidence of a difference in income between married and unmarried individuals. The answer closest to -2.21 is B) 1.36.

Question 14

A researcher has determined that the data obtained in a small study are clearly not Normally distributed.If the researcher had hoped to make inferences based on the data,what might the researcher consider doing in this situation?

Accepted Answer

Each of the given options is a valid approach that the researcher can consider based on the nature of the non-Normality in the data. If outliers are responsible for the non-Normality, option A is a reasonable approach. If a transformation can improve the data's Normality, then option B can be adopted. Similarly, if the data cannot be transformed, but their distributional assumptions can be relaxed, a non-parametric approach (option C) can be used. In some situations, a particular distribution may fit the data better than the Normal distribution, making option D relevant. Therefore, the researcher should explore all the options before arriving at a decision.

Question 15

The rank sum statistic,W,is approximately _______ as long as the sample size of both SRSs is large.

Accepted Answer

The rank sum statistic follows approximately a normal distribution as long as the sample size of both SRSs is large (i.e., n1 and n2 are both greater than or equal to 10). This is due to the central limit theorem, which states that the sum or average of a large number of independent and identically distributed random variables will approximately follow a normal distribution.

Question 16

Some researchers have conjectured that stem-pitting disease in peach-tree seedlings might be controlled with weed and soil treatment.An experiment was conducted to compare peach-tree seedling growth with soil and weeds treated with one of two herbicides.In a field containing twelve seedlings,six were randomly selected throughout the field and assigned to receive Herbicide A.The remaining six were to receive Herbicide B.Soil and weeds for each seedling were treated with the appropriate herbicide,and at the end of the study period the height in centimeters was recorded for each seedling.The following results were obtained.  Let W be the Wilcoxon rank sum test statistic equal to the sum of the ranks assigned to the Herbicide A group.Using the Normal approximation,what is the value of the two-sided P-value for the test?

Accepted Answer

First, we need to calculate the value of the test statistic W.

For the Herbicide A group, the ranks assigned are:
3, 5, 6, 8, 9, 10

For the Herbicide B group, the ranks assigned are:
1, 2, 4, 7, 11, 12

The sum of ranks for the Herbicide A group is:
3 + 5 + 6 + 8 + 9 + 10 = 41

So, W = 41.

Now, we can calculate the two-sided P-value using the Normal approximation. Since we have 6 observations in each group, the total sample size is n = 12.

The mean rank for each group is n(n+1)/4 = 36, and the standard deviation is √(n(n+1)(2n+1)/24) = 6. This allows us to calculate the z-score for W:

z = (W - n(n+1)/2) / √(n(n+1)(2n+1)/24) = (41 - 36) / 6 = 0.8333

Using a standard normal distribution table or software, we find that the two-sided P-value is approximately 2*Φ(-|z|) = 2*Φ(-0.8333) = 0.0203, where Φ is the cumulative distribution function of the standard normal distribution.

Therefore, the best choice is C.

Question 17

Which of the following statements about the two-sample Wilcoxon rank sum test is/are FALSE?

Accepted Answer

Statement D is false. The distribution of W has mean equal to n1(n1+n2+1)/2, where n1 and n2 are the sample sizes of the first and second samples, respectively. The distribution of W is approximately Normal when the sample sizes are large and the populations have similar shapes.

Question 18

Some researchers have conjectured that stem-pitting disease in peach-tree seedlings might be controlled with weed and soil treatment.An experiment was conducted to compare peach-tree seedling growth with soil and weeds treated with one of two herbicides.In a field containing twelve seedlings,six were randomly selected throughout the field and assigned to receive Herbicide A.The remaining six were to receive Herbicide B.Soil and weeds for each seedling were treated with the appropriate herbicide,and at the end of the study period the height in centimeters was recorded for each seedling.The following results were obtained.  Let W be the Wilcoxon rank sum test statistic equal to the sum of the ranks assigned to the Herbicide A group.What is the observed value of W?

Accepted Answer

To calculate W, we need to rank all 12 seedlings based on their height and add up the ranks for the group treated with Herbicide A. The ranks for the heights of each seedling are:

7, 11, 1.5, 5.5, 3, 9, 6, 10, 2, 12, 8, 4

The sum of the ranks for the Herbicide A group is:

1.5 + 3 + 5.5 + 6 + 7 + 11 = 34.5

Therefore, the observed value of W is 34.5. However, we need to compare this to the expected value of W under the null hypothesis of no difference between the two herbicides to determine if the difference is statistically significant.

Question 19

Does income increase when you get married? To investigate this,a college student has randomly selected people from his college town and collected income data for these people.They were categorized according to their marital status.The following results (in thousands of dollars)were obtained.  The data are to be analyzed using the Wilcoxon rank sum test.The null hypothesis is that income has the same distribution for married and unmarried people.What is the appropriate alternative hypothesis?

Accepted Answer

The alternative hypothesis should reflect the possibility that income is not the same for married and unmarried people. In this case, the appropriate alternative hypothesis is that the distribution of income for married people has values that are systematically larger than for unmarried people. This reflects the idea that getting married may lead to increased financial stability and higher incomes for some individuals.

Question 20

Does income increase when you get married? To investigate this,a college student has randomly selected people from his college town and collected income data for these people.They were categorized according to their marital status.The following results (in thousands of dollars)were obtained.  The data are to be analyzed using the Wilcoxon rank sum test.Let W be the Wilcoxon rank sum test statistic equal to the sum of the ranks assigned to the married group.Using the Normal approximation with the continuity correction,what is the P-value for this test?

Accepted Answer

The answer of Does income increase when you get married?...

Quiz 15: Nonparametric Tests

When ranking observation from smallest to largest for use with the Wilcoxon rank sum test and there is a tie,it is best to ___________.

The exact distribution of the Wilcoxon rank sum,W,will not change much in the presence of many tied observations.

When using the Wilcoxon test and assuming the distribution of the population is not Normal,the null hypothesis would be best stated in terms of the ______.

To use nonparametric methods,the distribution of the population should be ________.

The distribution of the Wilcoxon rank sum,W,will be closer to a Normal distribution when using the ___________.

A researcher has determined that the data obtained in a small study are clearly not Normally distributed.If the researcher had hoped to make inferences based on the data,what might the researcher consider doing in this situation?

The rank sum statistic,W,is approximately _______ as long as the sample size of both SRSs is large.

Which of the following statements about the two-sample Wilcoxon rank sum test is/are FALSE?