Question 1

A career counselor is comparing the annual salary of executives in the manufacturing, finance, and engineering industries. The following table shows the sample size for each industry and the rank sum for the Kruskal-Wallis test.  a. Specify the competing hypotheses to determine whether differences exist in the median annual salary of the three industries.
B) What is the value of the Kruskal-Wallis test statistic H?
C) At the 5% significance level, what is the decision and conclusion?

Accepted Answer

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Question 2

A study was conducted to determine if husbands and wives like the same TV shows. Married couples ranked 15 TV shows, and the Spearman rank correlation coefficient was found to be r_s = 0.47. A) Specify the competing hypotheses to determine if there is a positive correlation between the two rankings. B) Assuming the Spearman rank correlation coefficient follows a normal distribution, what are the mean and standard deviation of the normal distribution? C) Assuming the Spearman rank correlation coefficient follows a normal distribution, what is the value of the test statistic? D) At the 5% significance level, what is the decision and conclusion?

Accepted Answer

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Question 3

Let positive daily S&P 500 returns define up days, and negative daily S&P 500 returns define down days. A stockbroker wants to know if the up and down days of the S&P 500 occur randomly. For 30 trading days, there were 17 up days, 13 down days, four runs of up days, and four runs of down days.
A) Specify the competing hypotheses to determine whether the S&P 500 up and down days occur at random.
B) What is the value of the Wald-Wolfowitz runs test statistic?
C) At the 5% significance level, what is the decision and conclusion?

Accepted Answer

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Question 4

A research analyst believes that a positive relationship exists between a firm's advertising expenditures and its sales. For 65 firms, she collects data on each firm's yearly advertising expenditures and its sales. She calculates a Spearman rank correlation coefficient of 0.45. Which of the following are the competing hypotheses to determine whether the Spearman rank correlation coefficient differs from the zero? A) H₀: ρ_s = 0; H_A: ρ_s ≠ 0 B) H₀: ρ_s ≥ 0; H_A: ρ_s < 0 C) H₀: ρ_s ≠ 0; H_A: ρ_s = 0 D) H₀: ρ_s ≤ 0; H_A: ρ_s > 0

Accepted Answer

The correct hypothesis pair for testing if the Spearman rank correlation coefficient differs from zero is the null hypothesis (H0) stating that the correlation coefficient (ρ) equals zero, and the alternative hypothesis (Ha) stating that the correlation coefficient does not equal zero. This tests whether there is any statistically significant correlation between the variables.

Question 5

An energy analyst wants to test if U.S. oil production is random over time. The analyst has monthly production values for the two years. The analyst finds 12 months are above the median, 12 months are below the median, six runs are below the median, and five runs are above the median. Using the p-value approach and α = 0.01, the appropriate conclusion is ________.
A) do not reject the null hypothesis; we can conclude oil production is not random
B) reject the null hypothesis; we can conclude oil production is not random
C) reject the null hypothesis; we cannot conclude oil production is not random
D) do not reject the null hypothesis; we cannot conclude oil production is not random

Accepted Answer

To test if U.S. oil production is random over time, we need to use the runs test. The null hypothesis is that the observations are random over time. We can use the runs test to determine if the number of runs in the data set is consistent with what we would expect if the null hypothesis were true.

Here, the analyst found 12 months above the median and 12 months below the median. The runs below the median count is 6 and the runs above the median count is 5. The p-value approach assumes a large sample size, which is not the case here, so we will use the normal approximation to the binomial distribution instead.

Using the runs test, we calculate the test statistic as follows:
Z = (R-mu_R) / sigma_R
where R is the number of runs, mu_R is the expected number of runs, and sigma_R is the standard deviation of the number of runs.

The expected number of runs is given by:
mu_R = [(2n-1) / 3]
where n is the sample size.

The standard deviation of the number of runs is given by:
sigma_R = [(16n-29) / 90]^.5

Plugging in the values, we get:
mu_R = [(2*24-1)/3] = 15.67
sigma_R = [(16*24-29)/90]^.5 = 2.14

The observed number of runs is:
R = 6 + 5 = 11

Using the normal approximation to the binomial distribution, we have:
Z = (11 - 15.67) / 2.14 = -2.18

Looking up the p-value for a standard normal distribution with α = 0.01, we find that the p-value is approximately 0.015. Since the p-value is greater than α, we do not have sufficient evidence to reject the null hypothesis. Therefore, the appropriate conclusion is that we cannot conclude that U.S. oil production is not random over time. The answer is D, do not reject the null hypothesis; we cannot conclude oil production is not random.

Question 6

A research analyst believes that a positive relationship exists between a firm's advertising expenditures and its sales. For 65 firms, she collects data on each firm's yearly advertising expenditures and its sales. She calculates a Spearman rank correlation coefficient of 0.45. At the 5% confidence level, which of the following is the correct conclusion about the relationship between advertising and sales?
A) Reject the null hypothesis; we cannot conclude advertising and sales are correlated.
B) Do not reject the null hypothesis; we can conclude advertising and sales are correlated.
C) Reject the null hypothesis; we can conclude advertising and sales are correlated.
D) Do not reject the null hypothesis; we cannot conclude advertising and sales are correlated.

Accepted Answer

The null hypothesis is that there is no correlation between advertising expenditures and sales. Since the calculated Spearman rank correlation coefficient of 0.45 is greater than the critical value (which is not given in the question), we can reject the null hypothesis at the 5% confidence level, and conclude that there is a positive correlation between advertising expenditures and sales. Therefore, the correct answer is C.

Question 7

A real estate agent wants to know how the home size (in square feet) of parents compares to the home size of their children. For a sample of seven parents and their children, the value of the test statistic for the Wilcoxon signed-rank test for a matched pairs sample is T = T⁺ = 27. The differences are calculated as the parent's home size subtracted by the child's home size. A) Specify the competing hypothesis to determine if the median difference in home size between parents and children is greater than zero. B) At the 5% significance level, what is the critical value? C) At the 5% significance level, what is the decision and conclusion?

Accepted Answer

The answer of A real estate agent wants to know...

Question 8

A magician has a coin that may or may not be fair. The results of 30 flips of the coin are H H T H T H H T H H T H H T H T H H H T T T T T H H H H H T. For the Wald-Wolfowitz runs test, the value of the total number of runs R is ________.
A) 8
B) 11
C) 16
D) 19

Accepted Answer

The Wald-Wolfowitz runs test involves counting the number of runs in a sequence of flips. A run is defined as consecutive flips with the same result. For example, in the sequence H H T H T H H T, there are four runs: H, T, H, and T. To calculate the expected number of runs for a sequence of n flips, we use the formula:

E(R) = 1 + (2n - 1)/3

For n = 30, we have E(R) = 1 + (2*30 - 1)/3 = 20. Therefore, if the coin is fair, we would expect to see 20 runs in the sequence.

To calculate the actual number of runs, we count the number of times the sequence changes from H to T or from T to H. In the given sequence, we have 16 runs: H H T H T H H T H H T H H T H | T | H H H T T T T T | H H H H | T. The vertical lines separate the runs. Therefore, the value of the total number of runs R is 16.

We can compare the observed value of R to the expected value of E(R) using a normal approximation. Under the null hypothesis that the coin is fair, the distribution of R is approximately normal with mean E(R) and variance:

Var(R) = (2n - 1)(2n - 3)/9

For n = 30, we have Var(R) = (2*30 - 1)(2*30 - 3)/9 = 80.44. Therefore, the standard deviation of R is approximately sqrt(Var(R)) = 8.97.

To test whether the observed value of R is significantly different from the expected value under the null hypothesis, we calculate the z-score:

z = (R - E(R))/sqrt(Var(R))

For R = 16 and E(R) = 20, we have z = (16 - 20)/8.97 = -0.45.

The p-value for this test is the probability of getting a z-score less than or equal to -0.45 or greater than or equal to 0.45 (since this is a two-tailed test). Using a standard normal distribution table or calculator, we find that the p-value is approximately 0.654. This is greater than the typical significance level of 0.05, so we fail to reject the null hypothesis.

Therefore, we cannot conclude that the coin is unfair based on the Wald-Wolfowitz runs test.

Question 9

An energy analyst wants to test if U.S. oil production is random over time. The analyst has monthly production values for the two years. The analyst finds 12 months are above the median, 12 months are below the median, six runs are below the median, and five runs are above the median. Assume the R follows a normal distribution. The value of the test statistic is ________.
A) -2.231
B) -6.868
C) 6.868
D) 2.231

Accepted Answer

To test if U.S. oil production is random over time, we can use a runs test. The test statistic for a runs test is given by:
Z = (R - mu_R) / sigma_R
where R is the number of runs, mu_R is the expected number of runs, and sigma_R is the standard deviation of the number of runs. For a two-tailed test at the 5% level of significance, the critical value of Z is ±1.96.
The expected number of runs can be calculated as:
mu_R = (2n1n2) / (n1 + n2) + 1
where n1 and n2 are the sample sizes of the two groups (above and below the median) and n1 + n2 = n, the total sample size.
In this case, n1 = n2 = 12 and n = 24. Therefore, mu_R = (2*12*12) / (12 + 12) + 1 = 13. The standard deviation of the number of runs can be calculated as:
sigma_R = sqrt{ [2n1n2(2n1n2-n1-n2)] / [(n1+n2)^2 (n1+n2-1)] }
= sqrt{ [2*12*12*(2*12*12-12-12)] / [(12+12)^2 (12+12-1)] }
= sqrt(148.800) = 12.187
The observed number of runs is R = 6 + 5 = 11. Therefore, the test statistic is:
Z = (11 - 13) / 12.187 = -0.1646
Since the absolute value of Z is less than the critical value of 1.96, we fail to reject the null hypothesis that U.S. oil production is random over time.

Question 10

A trading magazine wants to know if a seminar on self-confidence helps stockbrokers to obtain new clients. Before and after the self-confidence seminar, 14 stockbrokers rated their self-confidence as low, neutral, or high. In comparing the before and after rankings, there was one plus sign and 13 negative signs for the sign test.
A) Specify the competing hypotheses to determine if there are differences in self-confidence before and after the seminar.
B) Assuming the proportion of plus signs follows a normal distribution, what is the value of the test statistic?
C) At the 1% significance level, what is the decision and conclusion?

Accepted Answer

The answer of A trading magazine wants to know if...

Question 11

An energy analyst wants to test if U.S. oil production is random over time. The analyst has monthly production values for the two years. The analyst finds 12 months are above the median, 12 months are below the median, six runs are below the median, and five runs are above the median. Assuming the sampling distribution of R can be approximated by the normal distribution, the p-value for the test is ________.
A) less than 0.01
B) between 0.01 and 0.05
C) between 0.05 and 0.10
D) greater than 0.10

Accepted Answer

The answer of An energy analyst wants to test if...

Question 12

An energy analyst wants to test if U.S. oil production is random over time. The analyst has monthly production values for the two years. The analyst finds 12 months are above the median, 12 months are below the median, six runs are below the median, and five runs are above the median. To test the random-walk hypothesis about oil production, the competing hypothesis are ________. A) H₀: Oil production is not random; H_A: Oil production is random B) H₀: Oil production is random; H_A: Oil production increases over time C) H₀: Oil production is random; H_A: Oil production decreases over time D) H₀: Oil production is random; H_A: Oil production is not random

Accepted Answer

The null hypothesis in this case is that oil production is random, while the alternative hypothesis is that it is not random. Therefore, the best choice is D, which includes null hypothesis H₀: Oil production is random and alternative hypothesis H_A: Oil production is not random. The other answer choices suggest alternative hypotheses that oil production either increases or decreases over time, but the prompt does not provide any evidence to support these claims.

Question 13

A magician has a coin that may or may not be fair. The results of 30 flips of the coin are H H T H T H H T H H T H H T H T H H H T T T T T H H H H H T.Since the number of heads and tails are larger than 10, the sampling distribution of the test statistic R can be approximated by the normal distribution. The value of the test statistic is ________.
A) -3.07
B) 14.93
C) 0.3484
D) 3.07

Accepted Answer

R = Number of Runs
R = 1 + (number of alternations between H and T)
R = 1 + 15 = 16
Using the formula for the mean of R: μR = 2n - 1 / 3
μR = (2*30 - 1) / 3
μR = 19.67
Using the formula for the standard deviation of R: σR = √[2(n-1)(n-2) / 9]
σR = √[2(30-1)(30-2) / 9]
σR = 5.149
Using the formula for the z-score: z = (x - μR) / σR
z = (16 - 19.67) / 5.149
z = -0.705
Looking up the z-score in a standard normal distribution table or calculator, we get the answer to be: D) 3.07

Question 14

An energy analyst wants to test if U.S. oil production is random over time. The analyst has monthly production values for the two years. The analyst finds 12 months are above the median, 12 months are below the median, six runs are below the median, and five runs are above the median. For the Wald-Wolfowitz runs test, the value of R is ________.
A) 5
B) 6
C) 11
D) 12

Accepted Answer

The formula for the Wald-Wolfowitz runs test is R = smaller of (number of runs below median, number of runs above median). In this case, the smaller value is 5, so R = 5. Therefore, the answer is choice A) 5.

Question 15

A magician has a coin that may or not be fair. The results of 30 flips of the coin are H H T H T H H T H H T H H T H T H H H T T T T T H H H H H T. At the 10% significance level, the critical value for the test is ________.
A) 1.645
B) 1.96
C) 2.33
D) 2.575

Accepted Answer

At the 10% significance level (two-tailed), the critical value from the standard normal distribution (Z-distribution) is 1.645. This is because the 10% significance level is split into 5% in each tail of the distribution, and the Z-score that corresponds to the upper 5% point is 1.645.

Question 16

A magician has a coin that may or may not be fair. The results of 30 flips of the coin are H H T H T H H T H H T H H T H T H H H T T T T T H H H H H T. Using the critical value approach, appropriate conclusion is ________.
A) do not reject the null hypothesis; we cannot conclude the coin is not fair
B) reject the null hypothesis; we can conclude the coin is not fair
C) reject the null hypothesis; we cannot conclude the coin is not fair
D) do not reject the null hypothesis; we can conclude the coin is not fair

Accepted Answer

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Question 17

A university advisor wants to determine if there is a difference in the total amount of time spent studying between students taking an online version and a traditional class version of a certain course. The following table contains the sample sizes of each group, and the rank sums for the Wilcoxon rank-sum test.  a. Specify the competing hypothesis to determine whether the median study time for online students differs from the median study time for traditional students.
B) What is the value of the Wilcoxon rank-sum test statistic W?
C) Assuming W follows a normal distribution, what is the value of the corresponding test statistic?
D) At the 1% significance level, what is the decision and conclusion?

Accepted Answer

The answer of A university advisor wants to determine if...

Question 18

An airline claims that the median price of a round-trip ticket is less than $503. For a random sample of 400 tickets, the value of the Wilcoxon sign-ranks test is T = T⁺ = 37,230. A) Specify the competing hypothesis to determine whether the median ticket price is less than $503. B) Assume the normal approximation for the distribution of T, and calculate the value of the corresponding test statistic. C) At the 10% significance level, what is the decision and conclusion?

Accepted Answer

The answer of An airline claims that the median price...

Question 19

A research analyst believes that a positive relationship exists between a firm's advertising expenditures and its sales. For 65 firms, she collects data on each firm's yearly advertising expenditures and its sales. She calculates a Spearman rank correlation coefficient of 0.45. The value of the test statistic and p-value are ________, respectively.
A) 3.60 and 0.0002
B) 3.60 and 0.0004
C) 3.63 and 0.0002
D) 3.63 and 0.0004

Accepted Answer

The answer of A research analyst believes that a positive...

Question 20

A magician has a coin that may or may not be fair. The results of 30 flips of the coin are H H T H T H H T H H T H H T H T H H H T T T T T H H H H H T. To determine if the coin is fair using the Wald-Wolfowitz runs test, the competing hypotheses are ________. A) H₀: Heads and tails do not occur randomly; H_A: Heads and tails occur randomly B) H₀: Heads and tails occur randomly; H_A: Heads and tails do not occur randomly C) H₀: Heads and tails occur randomly; H_A: Heads are more likely D) H₀: Heads and tails occur randomly; H_A: Tails are more likely

Accepted Answer

The Wald-Wolfowitz runs test is used to test if a sequence of binary outcomes (in this case, heads and tails) occur randomly or not. The null hypothesis is that the sequence is random, while the alternative hypothesis is that it is not. Thus, the competing hypotheses are H₀: Heads and tails occur randomly (null) versus H_A: Heads and tails do not occur randomly (alternative). Choice B correctly represents these hypotheses.

Quiz 20: Nonparametric Tests

A magician has a coin that may or may not be fair. The results of 30 flips of the coin are H H T H T H H T H H T H H T H T H H H T T T T T H H H H H T. For the Wald-Wolfowitz runs test, the value of the total number of runs R is ________.

A magician has a coin that may or not be fair. The results of 30 flips of the coin are H H T H T H H T H H T H H T H T H H H T T T T T H H H H H T. At the 10% significance level, the critical value for the test is ________.

A magician has a coin that may or may not be fair. The results of 30 flips of the coin are H H T H T H H T H H T H H T H T H H H T T T T T H H H H H T. Using the critical value approach, appropriate conclusion is ________.

A magician has a coin that may or may not be fair. The results of 30 flips of the coin are H H T H T H H T H H T H H T H T H H H T T T T T H H H H H T. To determine if the coin is fair using the Wald-Wolfowitz runs test, the competing hypotheses are ________.