Question 1

When we are comparing two populations and our two samples are such that each element of one sample relates to a particular element of the other sample, an appropriate nonparametric test would be the
A) one-sample sign test.
B) signed-rank test.
C) Wilcoxian test (rank sums).
D) U test.

Accepted Answer

The signed-rank test, specifically the Wilcoxon signed-rank test, is appropriate for comparing two related samples, where each element of one sample is paired with an element from the other sample. This test is used for nonparametric data, which means it doesn't assume a normal distribution of the differences between pairs.

Question 2

The nonparametric alternative to the two-sample $t$ test for the difference between two independent means is the
A) one-sample sign test.
B) paired-sample sign test.
C) U test.
D) Kruskal-Wallis test.

Accepted Answer

The nonparametric alternative to the two-sample $t$ test for the difference between two independent means is the U test, also known as the Mann-Whitney U test. This test is used when the assumptions of the $t$ test are not met, particularly regarding the normality of the distributions.

Question 3

The nonparametric alternative to the one-way analysis of variance is the
A) U test.
B) paired-sample sign test.
C) Wilcoxian test (rank sums).
D) $\mathrm{H}$ test.

Accepted Answer

The nonparametric alternative to the one-way analysis of variance (ANOVA) is the Kruskal-Wallis H test, which is used when the assumptions of ANOVA are not met, particularly regarding the normality of the data or when dealing with ordinal data.

Question 4

The Kruskal-Wallis test involves an application of the
A) normal distribution.
B) chi-square distribution.
C) binomial distribution.
D) $t$ distribution.

Accepted Answer

The Kruskal-Wallis test uses the chi-square distribution to determine if there are statistically significant differences between two or more groups of an independent variable on a continuous or ordinal dependent variable.

Question 5

The sign test applied to small samples of paired data requires the use of the
A) binomial distribution.
B) normal distribution.
C) $t$ distribution.
D) chi-square distribution.

Accepted Answer

The sign test uses the binomial distribution because it is a non-parametric test that compares two related samples, or repeated measurements on a single sample, to assess whether their population median differs. It is based on the direction of the differences between pairs, not their magnitude, making the binomial distribution appropriate for calculating probabilities of observed outcomes under the null hypothesis.

Question 6

Suppose that the $U$ test is to be applied to the following sequence.  The correct rankings in order are
A) 123456
B) 12.52 .5456
C) 1234.54 .56
D) 12.52 .54 .54 .56

Accepted Answer

The answer of Suppose that the $U$ test is to...

Question 7

The sign test is used with $n=9$ to test the null hypothesis that the new method and the old method produce equally high scores against the alternative hypothesis that the new method produces higher scores than the old. If we obtain eight plus signs in subtracting (new score - old score), then with $\alpha=0.05$, the null hypothesis should
A) be rejected since $6<7$.
B) not be rejected since $6<7$.
C) be rejected since $7<8$.
D) not be rejected since $7<8$.

Accepted Answer

The sign test is a non-parametric test used to determine if there are more positive differences than would be expected by chance if the two methods were equally good. With $n=9$ and eight plus signs, we are looking at a one-tailed test (since we're testing if the new method is better). The critical value for $\alpha=0.05$ with $n=9$ in a one-tailed test is 7 (meaning we would expect 7 or fewer plus signs by chance if the null hypothesis were true). Since we have 8 plus signs, which is greater than 7, we reject the null hypothesis in favor of the alternative that the new method produces higher scores.

Question 8

The Mann-Whitney test is the same as the
A) one-sample sign test.
B) U test.
C) Kruskal-Wallis test.
D) paired-sample sign test.

Accepted Answer

The Mann-Whitney test is also known as the U test. It is a non-parametric test used to compare differences between two independent groups on a single, ordinal or continuous variable.

Question 9

Which of the following tests must be two-tailed?
A) U test
B) one-sample sign test
C) number of runs test
D) paired-sample sign test

Accepted Answer

The number of runs test inherently requires a two-tailed approach because it is designed to determine if a sequence of observations is too random or not random enough, implying deviations in either direction from the null hypothesis.

Question 10

If the $U$ test is applied in a test whose alternative hypothesis is $\mu_{1} \neq \mu_{2}$ with $n_{1}=7, n_{2}=7, \alpha=0.05$, the null hypothesis can be rejected if
A) $W_{1} \geq 70$ and $W_{2} \geq 70$.
B) $\mathrm{U} \leq 8$.
C) $\mathrm{U}>8$.
D) $\mathrm{W}_{1} \leq 70$ or $\mathrm{W}_{2} \leq 70$.

Accepted Answer

The Mann-Whitney U test, often referred to as the U test, is used to compare differences between two independent groups when the dependent variable is either ordinal or continuous, but not normally distributed. The critical value of U for rejecting the null hypothesis depends on the sample sizes of the two groups and the significance level ($\alpha$). Given $n_{1} = n_{2} = 7$ and $\alpha = 0.05$, the critical value of U can be looked up in a table specific to the Mann-Whitney U test. For these sample sizes and a two-tailed test (since the alternative hypothesis is $\mu_{1} \neq \mu_{2}$), the critical value of U to reject the null hypothesis is indeed less than or equal to 8. This means if the calculated U statistic is 8 or less, the null hypothesis can be rejected, indicating a significant difference between the two groups.

Question 11

Independent random samples of eight males and seven female students at a college showed the following weekly food expenses in dollars.
Men: $20,21,25,28,30,28,24,26$
Female: $25,18,17,23,20,24,22$
Test the null hypothesis that the men and women have the same average food expenses against the alternative hypothesis that they do not have the same average food expenses. Use $\alpha=0.05$.

Accepted Answer

The answer of Independent random samples of eight males and...

Question 12

The following are monthly sales (in thousands of dollars) of two department stores, A and B, for the months January through December 1979:
10 and 9,7 and 4,8 and 5, 7 and 6,11 and 10,15 and 16,
15 and 13,12 and 13,9 and 8,18 and 17,19 and 13,21 and 20.
-For the situation above, use the small sample sign test with $\alpha=0.05$ to test the null hypothesis that on the average the two department stores have equal sales against the alternative that on the average
a. department store $\mathrm{A}$ has more sales.
b. there is a difference in sales between the two stores.

Accepted Answer

The answer of The following are monthly sales (in thousands...

Question 13

The following are monthly sales (in thousands of dollars) of two department stores, A and B, for the months January through December 1979:
10 and 9,7 and 4,8 and 5, 7 and 6,11 and 10,15 and 16,
15 and 13,12 and 13,9 and 8,18 and 17,19 and 13,21 and 20.
-For the situation above, use the large sample sign test with $\alpha=0.05$ to test the null hypothesis that on the average the two department stores have equal sales against the alternative that on the average
a. department store $\mathrm{A}$ has more sales.
b. there is a difference in sales between the two stores.

Accepted Answer

The answer of The following are monthly sales (in thousands...

Question 14

Students were observed as they arrived at a college dance. The following is the order in which men and women arrived at the dance:
$M M M M M W M M W W$
$M M M W M W W M M M$
Test the sequence for randomness at the 0.05 level of significance.

Accepted Answer

The answer of Students were observed as they arrived at...

Question 15

A test was conducted to compare the miles per gallon obtained by three sub-compact automobiles. The following data (in miles per gallon) were obtained from 6 full tanks of the same gasoline.
I. $24,22,23,25,22,21$
II. $23,22,26,24,27,25$
III. $30,28,26,27,26,24$
Use a nonparametric test to evaluate whether or not there are significant differences in gasoline economy among the automobiles. Use $\alpha=0.05$.

Accepted Answer

The answer of A test was conducted to compare the...

Question 16

Random samples of 15 Model I cars and 13 Model II cars were obtained and the number of miles (in thousands) that each lasted from new until the car was "junked" was recorded.
Model I: $\quad 70,60,80,75,81,73,80,90,82,93,89,95,91,88,90$
Model II: $\quad 88,80,95,105,90,99,93,88,95,102,98,95,91$
Test the null hypothesis that the two models have the same average life expectancy against the alternative hypothesis that they do not have the same average life expectancy. Use $\alpha=0.01$.

Accepted Answer

The answer of Random samples of 15 Model I cars...

Question 17

An aptitude test is given to trainees on a high-tech form. Here are the grades on the test in the order that the trainees completed the test:
$98,92,95,85,80,78,75,83,82,95,92,87,80,65,92,93,82,98,93,72,70$.
Test whether the trainees finished the test in random order with regard to grades. Use $\alpha=0.01$.

Accepted Answer

The answer of An aptitude test is given to trainees...

Question 18

Table 18.1
 
-The mathematics S.A.T. scores and college grade point averages of eight students are given in table 18.1.
a. Calculate the rank-correlation coefficient $r_{S}$.
b. Test the null hypothesis of no correlation using $r_{\mathrm{S}}$. Let $\alpha=0.05$.

Accepted Answer

The answer of Table 18.1
 
-The mathematics S.A.T. scores and...

Question 19

Table 18.2
 
-Table 18.2 shows the scores made on the verbal and mathematical components of an S.A.T. by 15 high -school seniors who plan to go to a certain college. Use the table to calculate the Spearman Rank-Correlation Coefficient for the test scores.

Accepted Answer

The answer of Table 18.2
 
-Table 18.2 shows the scores...

Question 20

Table 18.3
A consumer advocate is interested in testing whether the number of french fries in a portion in a particular cafeteria is different from that of a particular restaurant. The table below shows the results when random samples of 10 portions of french fries are selected and fries counted.
 
-Use Table 18.3 to conduct the test using $\alpha=0.01$.

Accepted Answer

The answer of Table 18.3
A consumer advocate is interested in...

Quiz 18: Nonparametric Tests

When we are comparing two populations and our two samples are such that each element of one sample relates to a particular element of the other sample, an appropriate nonparametric test would be the

The nonparametric alternative to the two-sample $t$ test for the difference between two independent means is the

The nonparametric alternative to the one-way analysis of variance is the

The Kruskal-Wallis test involves an application of the

The sign test applied to small samples of paired data requires the use of the

Suppose that the $U$ test is to be applied to the following sequence. The correct rankings in order are

The Mann-Whitney test is the same as the

Which of the following tests must be two-tailed?

If the $U$ test is applied in a test whose alternative hypothesis is $\mu_{1} \neq \mu_{2}$ with $n_{1}=7, n_{2}=7, \alpha=0.05$ , the null hypothesis can be rejected if

Students were observed as they arrived at a college dance. The following is the order in which men and women arrived at the dance: $M M M M M W M M W W$ $M M M W M W W M M M$ Test the sequence for randomness at the 0.05 level of significance.

Table 18.2 -Table 18.2 shows the scores made on the verbal and mathematical components of an S.A.T. by 15 high -school seniors who plan to go to a certain college. Use the table to calculate the Spearman Rank-Correlation Coefficient for the test scores.