Question 1

Which of the following refers to a nonparametric method to compare two populations,when the samples are independent and the data are either ordinal or interval,and where the normality requirement necessary to perform the parametric test is not satisfied? A)Sign test B)Wilcoxon signed rank test C)Equal-variance t-test of μ₁ - μ₂ D)Wilcoxon rank sum test or Mann-Whitney test

Accepted Answer

The Wilcoxon rank sum test (also known as the Mann-Whitney test) is a nonparametric method used to compare two independent populations when the data are either ordinal or interval but do not meet the normality assumption needed for a parametric test. The other options, the sign test and Wilcoxon signed-rank test, are appropriate for paired data, not independent samples. The third option, the equal-variance t-test, assumes normality of the data and equal variances, so it is not appropriate for non-normal data.

Question 2

The number of degrees of freedom in a chi-square goodness-of-fit-test when population parameters are estimated is equal to the number of:
A)categories.
B)categories minus 1.
C)categories minus the number of parameters estimated.
D)categories minus the number of parameters estimated minus 1.

Accepted Answer

The degrees of freedom in a chi-square goodness-of-fit test is calculated as the number of categories minus the number of parameters estimated minus 1. This accounts for the constraints imposed by estimating parameters.

Question 3

In a goodness-of-fit-test,suppose that the value of the test statistic is 13.08 and degrees of freedom = 6.At the 5% significance level,the null hypothesis is:
A)rejected,and p-value for the test is smaller than 0.05.
B)not rejected,and p-value for the test is greater than 0.05.
C)rejected,and p-value for the test is greater than 0.05.
D)not rejected,and p-value for the test is smaller than 0.05.

Accepted Answer

At the 5% significance level with 6 degrees of freedom, the critical value for the test statistic is 12.59. Since the test statistic value of 13.08 is greater than the critical value of 12.59, we reject the null hypothesis. Moreover, the p-value for the test is smaller than 0.05, indicating strong evidence against the null hypothesis. Therefore, the correct choice is A.

Question 4

A chi-square goodness-of-fit-test is to be performed to see if data fit the Poisson distribution.There are 8 categories,and the parameter λ must be estimated.How many degrees of freedom should be used?
A)8
B)7
C)6
D)5

Accepted Answer

For a goodness-of-fit test with k categories and estimated parameter(s), the degrees of freedom is equal to (k-1) minus the number of estimated parameters. In this case, there are 8 categories and 1 estimated parameter (λ), so the degrees of freedom would be (8-1)-1=6. Therefore, the correct choice is C)6.

Question 5

A two-way table used for a test of independence is sometimes called a:
A)contingency table.
B)frequency table.
C)t table.
D)manifold table.

Accepted Answer

A two-way table used for a test of independence is commonly known as a contingency table. It shows the distribution of two categorical variables and is used to check for the dependence or independence between them. The other options, frequency table, t table, and manifold table, are not appropriate terms for a two-way table used for a test of independence.

Question 6

Which of the following should certainly be known to select a value in the chi-square distribution table?
A)the mean and the variance of the observations
B)the level of significance and the variance of the observations
C)the degrees of freedom and the mean of the observations
D)the level of significance and the degrees of freedom

Accepted Answer

In order to select a value in the chi-square distribution table, one needs to know the level of significance (usually denoted by alpha) and the degrees of freedom (typically denoted by df). The mean and variance of the observations are not necessary for determining values in the chi-square distribution table.

Question 7

A contingency table is to be used to test for independence.There are 3 rows and 3 columns in the table.How many degrees of freedom are associated with this test?
A)9
B)6
C)4
D)2

Accepted Answer

The degrees of freedom for a contingency table are calculated using the formula (number of rows - 1) x (number of columns - 1). In this case, (3-1) x (3-1) = 4, which means there are 4 degrees of freedom associated with the test. Therefore, the best choice is C.

Question 8

Consider a cell in a contingency table.Given the cell's row total of 40,the cell's column total of 50,and a sample size of 250,the cell's expected frequency:
A)equals 8.
B)equals 5.
C)equals 40.
D)cannot possibly be determined without additional information.

Accepted Answer

The expected frequency for a cell in a contingency table is calculated as (row total * column total) / sample size. Here, it would be (40 * 50) / 250 = 8.

Question 9

A chi-square goodness-of-fit-test is to be performed.If the degrees of freedom are 8,and α = 0.10,then the table chi-square value that defines the rejection region is:
A)13.362
B)15.507
C)17.535
D)12.017

Accepted Answer

The critical value for a chi-square goodness-of-fit test with 8 degrees of freedom and a significance level of 0.10 is 13.362.

Question 10

A test of independence is to be performed.The table has 4 rows and 5 columns.What would the degrees of freedom be?
A)20
B)15
C)12
D)9

Accepted Answer

The degrees of freedom for a test of independence are calculated as (number of rows - 1) x (number of columns - 1). So, in this case, (4-1) x (5-1) = 3 x 4 = 12, which is option C.

Question 11

In a normal approximation to the Wilcoxon signed rank test,the test statistic is calculated as z = 1.26.For a two-tail test,the p-value is:
A)0.2076
B)0.3962
C)0.1238
D)0.3974

Accepted Answer

In a normal approximation to the Wilcoxon signed-rank test, the test statistic is compared with a standard normal distribution to obtain the p-value. For a two-tailed test, the p-value is twice the area in the tails beyond the absolute value of the test statistic. In this case, the absolute value of the test statistic is 1.26, so the area in each tail beyond 1.26 is 0.1038. Therefore, the p-value is 2*0.1038 = 0.2076.

Question 12

A chi-square test of a contingency table with 4 rows and 5 columns shows that the value of the test statistic is 22.18.The most accurate statement that can be made about the p-value for this test is that the p-value is:
A)greater than 0.05 but smaller than 0.10.
B)smaller than 0.025.
C)greater than 0.025 but smaller than 0.05.
D)greater than 0.10.

Accepted Answer

Without knowing the degrees of freedom or significance level (α), we cannot determine the exact p-value. However, based on the test statistic value (22.18), it is likely that the p-value is smaller than 0.05. Since the question specifies that the most accurate statement that can be made about the p-value, the only option that fits is C, which states that the p-value is greater than 0.025 but smaller than 0.05. Option A is ruled out because the p-value cannot be greater than the significance level of 0.05. Option B is too specific and limits the range of the p-value too much. Option D is unlikely since the test statistic is relatively large.

Question 13

The significance level for a Wilcoxon signed rank test is 0.05.The alternative hypothesis is stated as: The location of population 1 is to the left of the location of population 2.The appropriate critical value for a sample of size 20 (that is the number of nonzero differences)is:
A)122
B)150
C)61
D)75

Accepted Answer

For a Wilcoxon signed rank test with a significance level of 0.05 and an alternative hypothesis stating that the location of population 1 is to the left of population 2, the appropriate critical value for a sample size of 20 (number of nonzero differences) is 61. This can be found in a table of critical values for the Wilcoxon signed rank test. The other answer choices are not appropriate critical values for this scenario.

Question 14

Which of the following statements about nonparametric tests is false?
A)We can safely employ nonparametric tests even when we know nothing at all about the population from which sample data are being drawn.
B)We can conduct nonparametric tests with nominal data.
C)We can conduct nonparametric tests with ordinal data.
D)We can only conduct nonparametric tests with data of a higher order than nominal or ordinal.

Accepted Answer

This statement is false. Nonparametric tests can be conducted with both nominal and ordinal data, and some tests can even be conducted with interval or ratio data.

Question 15

A chi-square goodness-of-fit-test is being used to determine if the observed frequencies and expected frequencies from five categories are significantly different.If only one parameter is estimated from the data,what is the critical chi-square value associated with α = 0.05?
A)3.84
B)5.99
C)7.81
D)9.49

Accepted Answer

The degrees of freedom for the test is calculated as the number of categories minus the number of estimated parameters minus one, which is 5 - 1 - 1 = 3. The critical chi-square value for 3 degrees of freedom at α = 0.05 is 7.81.

Question 16

A test of independence is to be performed.The table has 4 rows and 3 columns.What would the degrees of freedom be?
A)12
B)6
C)9
D)3

Accepted Answer

The degrees of freedom for a test of independence in a contingency table are calculated as (number of rows - 1) * (number of columns - 1). Here, it would be (4 - 1) * (3 - 1) = 3 * 2 = 6.

Question 17

The nonparametric tests like the Sign test,the Wilcoxon rank sum test,the Mann-Whitney test,and the Wilcoxon signed rank test require that the probability distributions be:
A)identical except with respect to location.
B)identical except with respect to spread (variance).
C)identical except with respect to shape (distribution).
D)different with respect to location,spread,and shape.

Accepted Answer

The nonparametric tests mentioned above require that the probability distributions be identical except with respect to location. This means that the shapes and spreads of the distributions can differ, but the center point must be the same. These tests are used to compare the medians of two or more groups, and they do not assume normality or equal variances, making them robust to violations of these assumptions.

Question 18

In the sign test for paired differences,the test statistic S has a:
A)binomial distribution with P = 0.5.
B)normal distribution.
C)chi-square distribution.
D)Poisson distribution.

Accepted Answer

The sign test for paired differences is a non-parametric test used to determine whether the median of the differences between two paired samples is zero or not. The test statistic S is the number of pairs where the second value is greater than the first value. Under the null hypothesis that the median difference is zero, S follows a binomial distribution with P = 0.5. Therefore, the best choice is A.

Question 19

For a random sample of 247 observations,the skewness of the data was calculated to be -0.52 while the kurtosis was calculated to be 3.24.Which of the following is true in the light of the information given above? A)Reject H₀ for α = 0.05,but not for α = 0.10 B)Reject H₀ for α = 0.10,but not for α = 0.5 C)Reject H₀ for α= 0.05 and for α = 0.10 D)Fail to reject H₀ for all levels of significance.

Accepted Answer

For a sample size of 247, the critical values for skewness and kurtosis at α = 0.05 are -1.96 and 3.29, respectively. Since the calculated skewness (-0.52) is within the critical values, we fail to reject H0 for skewness. However, the calculated kurtosis (3.24) is greater than the critical value, so we reject H0 for kurtosis. Therefore, the correct answer is C.

Question 20

A Wilcoxon rank sum test for comparing two populations involves two independent samples of sizes 15 and 20.The un-standardized test statistic (that is the rank sum)is T = 225.The value of the standardized test statistic z is:
A)6)0
B)15
C)6)43
D)-1.5

Accepted Answer

The standardized test statistic z is calculated using the formula: z = (T - E(T)) / sqrt(Var(T)), where E(T) is the expected value of the rank sum and Var(T) is the variance. For the Wilcoxon rank sum test, E(T) = n1(n1+n2+1)/2 and Var(T) = n1*n2(n1+n2+1)/12. Substituting the given values, we find z = -1.5.

Quiz 14: Analysis of Categorical Data

Which of the following refers to a nonparametric method to compare two populations,when the samples are independent and the data are either ordinal or interval,and where the normality requirement necessary to perform the parametric test is not satisfied?

The number of degrees of freedom in a chi-square goodness-of-fit-test when population parameters are estimated is equal to the number of:

In a goodness-of-fit-test,suppose that the value of the test statistic is 13.08 and degrees of freedom = 6.At the 5% significance level,the null hypothesis is:

A chi-square goodness-of-fit-test is to be performed to see if data fit the Poisson distribution.There are 8 categories,and the parameter λ must be estimated.How many degrees of freedom should be used?

A two-way table used for a test of independence is sometimes called a:

Which of the following should certainly be known to select a value in the chi-square distribution table?

A contingency table is to be used to test for independence.There are 3 rows and 3 columns in the table.How many degrees of freedom are associated with this test?

Consider a cell in a contingency table.Given the cell's row total of 40,the cell's column total of 50,and a sample size of 250,the cell's expected frequency:

A chi-square goodness-of-fit-test is to be performed.If the degrees of freedom are 8,and α = 0.10,then the table chi-square value that defines the rejection region is:

A test of independence is to be performed.The table has 4 rows and 5 columns.What would the degrees of freedom be?

In a normal approximation to the Wilcoxon signed rank test,the test statistic is calculated as z = 1.26.For a two-tail test,the p-value is:

A chi-square test of a contingency table with 4 rows and 5 columns shows that the value of the test statistic is 22.18.The most accurate statement that can be made about the p-value for this test is that the p-value is:

The significance level for a Wilcoxon signed rank test is 0.05.The alternative hypothesis is stated as: The location of population 1 is to the left of the location of population 2.The appropriate critical value for a sample of size 20 (that is the number of nonzero differences) is:

Which of the following statements about nonparametric tests is false?

A chi-square goodness-of-fit-test is being used to determine if the observed frequencies and expected frequencies from five categories are significantly different.If only one parameter is estimated from the data,what is the critical chi-square value associated with α = 0.05?

A test of independence is to be performed.The table has 4 rows and 3 columns.What would the degrees of freedom be?

The nonparametric tests like the Sign test,the Wilcoxon rank sum test,the Mann-Whitney test,and the Wilcoxon signed rank test require that the probability distributions be:

In the sign test for paired differences,the test statistic S has a:

For a random sample of 247 observations,the skewness of the data was calculated to be -0.52 while the kurtosis was calculated to be 3.24.Which of the following is true in the light of the information given above?

A Wilcoxon rank sum test for comparing two populations involves two independent samples of sizes 15 and 20.The un-standardized test statistic (that is the rank sum) is T = 225.The value of the standardized test statistic z is: