A coach records the sprint times of 12 athletes before and after a training program and wants to know whether times improved. Which procedure fits this design?
Paired-difference t test
✓
Two-sample t test for independent means
Two-proportion z test
One-proportion z test
Correct Answer
Paired-difference t test
Each athlete is measured twice, so the before and after times are linked and the analysis uses the difference for each athlete in a paired-difference t test. A two-sample t test for independent means would wrongly treat the before and after times as unrelated groups. A two-proportion z test compares percentages, but sprint times are numerical measurements. A one-proportion z test is about a single percentage, not about times.
Category: Choosing the Test
Question 2
Researchers randomly assign 30 patients to a new diet and a different 30 patients to a standard diet, then compare mean weight loss. The population standard deviations are unknown. Which procedure fits this design?
Matched pairs t test
Two-sample t test for independent means
✓
Two-proportion z test
Two-sample z test with known sigmas
Correct Answer
Two-sample t test for independent means
The two groups contain different, unrelated patients, the outcome is a numerical mean, and sigma is unknown, so a two-sample t test for independent means applies. A matched pairs t test needs each measurement in one group to be linked to a specific measurement in the other. A two-proportion z test compares percentages rather than mean weight loss. A two-sample z test with known sigmas is not possible here, because the population standard deviations are unknown.
Category: Choosing the Test
Question 3
A pollster takes separate random samples of voters in two states and records whether each voter supports a ballot measure. She wants to compare the levels of support in the two states. Which procedure fits this design?
Matched pairs t test
Two-sample t test for independent means
Two-proportion z test
✓
One-sample t test
Correct Answer
Two-proportion z test
Each response is yes or no, and the samples are independent, so the comparison is between two population proportions using a two-proportion z test. A matched pairs t test needs linked numerical measurements. A two-sample t test for independent means compares numerical averages, not shares of yes answers. A one-sample t test concerns a single mean, not two groups.
Category: Choosing the Test
Question 4
A researcher believes the mean commute time in City 1 is longer than the mean commute time in City 2. Which alternative hypothesis fits this belief?
Ha: mu1 < mu2
Ha: mu1 = mu2
Ha: x-bar1 > x-bar2
Ha: mu1 > mu2
✓
Correct Answer
Ha: mu1 > mu2
The claim is that the population mean for City 1 exceeds the population mean for City 2, which is Ha: mu1 > mu2, equivalent to mu1 - mu2 > 0. Ha: mu1 < mu2 reverses the direction of the claim. Ha: mu1 = mu2 cannot be an alternative, because the equality belongs in the null hypothesis. Ha: x-bar1 > x-bar2 describes the sample means, but hypotheses concern population parameters.
Category: Writing Hypotheses
Question 5
An analyst suspects that the proportion of customers who renew a subscription is lower for Plan A than for Plan B. Which alternative hypothesis fits this suspicion?
Ha: pA < pB
✓
Ha: pA > pB
Ha: pA = pB
Ha: p-hatA < p-hatB
Correct Answer
Ha: pA < pB
The suspicion is that the population renewal proportion for Plan A is below that for Plan B, written Ha: pA < pB. Ha: pA > pB points in the opposite direction. Ha: pA = pB is the null hypothesis of no difference, not an alternative. Ha: p-hatA < p-hatB is a statement about sample proportions, which are already known once the data are collected.
Category: Writing Hypotheses
Question 6
In a matched pairs test, the null hypothesis is written in terms of which parameter?
The difference of two independent sample means
The population mean of the differences, mu_d
✓
The difference of two population proportions
The sample standard deviation of the differences
Correct Answer
The population mean of the differences, mu_d
Pairing turns the data into a single list of differences, and the hypotheses concern the population mean of those differences, usually H0: mu_d = 0. The difference of two independent sample means belongs to an independent-samples test and is a statistic, not a parameter. The difference of two population proportions belongs to a two-proportion test. The sample standard deviation of the differences is used to compute the standard error, but it is not what the hypotheses describe.
Category: Matched Pairs
Question 7
Two independent random samples come from normal populations whose standard deviations are both known. Which distribution does the test statistic for the difference in means follow?
Student's t distribution
Binomial distribution
Standard normal distribution
✓
Uniform distribution
Correct Answer
Standard normal distribution
When both sigmas are known, the standardized difference x-bar1 - x-bar2 is a z-score that follows the standard normal distribution. Student's t distribution is used when the population standard deviations must be estimated from the samples. The binomial distribution describes counts of successes, not a difference of means. The uniform distribution is not the sampling distribution of a difference of means.
Category: Comparing Two Means
Question 8
In the Aspin-Welch t test for two independent means, which allows the population standard deviations to differ, how are the degrees of freedom found?
As the sum of the two sample sizes
As the larger sample size minus 1
As the number of pairs minus 1
From a formula using both sample variances and sizes
✓
Correct Answer
From a formula using both sample variances and sizes
The Aspin-Welch degrees of freedom come from a formula that combines s1^2/n1 and s2^2/n2 with both sample sizes, and the result is often not a whole number, so software or a calculator usually computes it. The sum of the two sample sizes is too large and ignores the variances. The larger sample size minus 1 is not the rule for this test and also ignores the variances. The number of pairs minus 1 is the rule for matched pairs, not for independent samples.
Category: Comparing Two Means
Question 9
Two independent samples give x-bar1 = 60, s1 = 9, n1 = 9 and x-bar2 = 50, s2 = 8, n2 = 4. For H0: mu1 - mu2 = 0, what is the t test statistic, using the standard error sqrt(s1^2/n1 + s2^2/n2)? Round to two decimal places.
2.00
✓
1.43
0.40
0.83
Correct Answer
2.00
The standard error is sqrt(81/9 + 64/4) = sqrt(9 + 16) = 5, so t = (60 - 50) / 5 = 2.00. The value 1.43 comes from adding the two separate standard errors, 3 + 4 = 7, instead of adding their squares. The value 0.40 comes from dividing by 25 and forgetting the square root. The value 0.83 comes from using sqrt(81 + 64) and ignoring the sample sizes.
Category: Comparing Two Means
Question 10
Two independent samples come from normal populations with sigma1 = 4, n1 = 16 and sigma2 = 6, n2 = 12. The sample means differ by x-bar1 - x-bar2 = 3. For H0: mu1 - mu2 = 0, what is the z test statistic? Round to two decimal places.
1.10
1.50
✓
0.75
3.00
Correct Answer
1.50
The standard error is sqrt(16/16 + 36/12) = sqrt(1 + 3) = 2, so z = 3 / 2 = 1.50. The value 1.10 comes from adding the separate standard errors, 1 + 1.732, instead of their squares. The value 0.75 divides by 4 and forgets the square root. The value 3.00 is the raw difference without dividing by the standard error.
Category: Comparing Two Means
Question 11
In Sample A, 40 of 100 people say yes. In Sample B, 60 of 300 people say yes. What is the pooled proportion used in a z test comparing the two proportions?
0.30
0.20
0.25
✓
0.60
Correct Answer
0.25
The pooled proportion combines all successes over all observations: (40 + 60) / (100 + 300) = 100 / 400 = 0.25. The value 0.30 is the simple average of 0.40 and 0.20, which ignores that Sample B is three times larger. The value 0.20 is the difference between the two sample proportions. The value 0.60 is the sum of the two sample proportions.
Category: Comparing Two Proportions
Question 12
In a z test of H0: pA = pB, why are the two samples combined into one pooled proportion for the standard error?
Pooling makes the two sample sizes equal
Pooling removes the need for random samples
Pooling turns the test into a matched pairs test
H0 says the two proportions are equal, so one shared value is estimated
✓
Correct Answer
H0 says the two proportions are equal, so one shared value is estimated
The test assumes H0 is true while computing the standard error, and under H0 both groups share a single population proportion, which is best estimated by pooling all the data. Pooling makes the two sample sizes equal is false, because the sample sizes stay as they were. Pooling removes the need for random samples is false, because both samples must still be random and independent. Pooling turns the test into a matched pairs test is false, because the samples remain independent.
Category: Comparing Two Proportions
Question 13
A matched pairs study has n = 16 pairs, a mean difference of x-bar_d = 3, and a standard deviation of the differences of s_d = 4. For H0: mu_d = 0, what is the t test statistic? Round to two decimal places.
3.00
✓
0.75
12.00
0.19
Correct Answer
3.00
The standard error of the mean difference is s_d / sqrt(n) = 4 / 4 = 1, so t = 3 / 1 = 3.00. The value 0.75 divides by s_d alone and forgets the sample size. The value 12.00 divides by s_d / n instead of s_d / sqrt(n). The value 0.19 divides the mean difference by n.
Category: Matched Pairs
Question 14
A matched pairs study measures 12 subjects twice each. How many degrees of freedom does the matched pairs t test have?
22
11
✓
23
12
Correct Answer
11
A matched pairs test is a one-sample t test on the 12 differences, so df = 12 - 1 = 11. The value 22 treats the data as two independent samples of 12 and uses 12 + 12 - 2. The value 23 counts all 24 measurements and subtracts 1. The value 12 forgets to subtract 1.
Category: Matched Pairs
Question 15
Why can a matched pairs design detect a treatment effect that a design with two independent groups might miss?
It doubles the number of subjects in the study
It makes the population standard deviation known
It removes variation caused by differences between subjects
✓
It guarantees that the data are normally distributed
Correct Answer
It removes variation caused by differences between subjects
Because each subject serves as its own comparison, natural differences between people cancel out in the differences, which usually leaves a smaller standard deviation and a more sensitive test. It doubles the number of subjects in the study is false, because the same subjects are measured twice. It makes the population standard deviation known is false, because the standard deviation of the differences is still estimated from the sample. It guarantees that the data are normally distributed is false, because pairing has no effect on the shape of the population.
Category: Matched Pairs
Question 16
Two groups have sample means of 75 and 70, and their pooled standard deviation is 10. What is Cohen's d? Round to two decimal places.
2.00
5.00
0.05
0.50
✓
Correct Answer
0.50
Cohen's d is the difference in means divided by the pooled standard deviation: (75 - 70) / 10 = 0.50. The value 2.00 divides the standard deviation by the difference, which is upside down. The value 5.00 is the raw difference in means without standardizing. The value 0.05 divides the difference by 100, which is the pooled variance rather than the pooled standard deviation.
Category: Effect Size
Question 17
A study reports Cohen's d = 0.9 for the difference between two group means. Using the usual benchmarks, how is this effect described?
Large effect
✓
Medium effect
Small effect
No effect
Correct Answer
Large effect
Cohen's conventional benchmarks are about 0.2 for a small effect, 0.5 for a medium effect and 0.8 for a large effect, so 0.9 counts as large. Medium effect fits values near 0.5. Small effect fits values near 0.2. No effect would mean d is near 0.
Category: Effect Size
Question 18
A z test comparing two population proportions at alpha = 0.05 gives a p-value of 0.08. Which conclusion is appropriate?
There is sufficient evidence that the population proportions differ
There is not sufficient evidence that the population proportions differ
✓
The population proportions are proven to be equal
The sample proportions must have been exactly equal
Correct Answer
There is not sufficient evidence that the population proportions differ
Since the p-value of 0.08 is larger than alpha, H0 is not rejected, so the data do not give enough evidence of a difference. There is sufficient evidence that the population proportions differ would require a p-value below 0.05. The population proportions are proven to be equal goes too far, because not rejecting H0 is not proof. The sample proportions must have been exactly equal is false, because a p-value of 0.08 can come from sample proportions that differ somewhat.
Category: Comparing Two Proportions
Question 19
A two-sample t test of Ha: mu1 not equal to mu2 at alpha = 0.01 gives a p-value of 0.002. What is the decision and conclusion?
Do not reject H0 because the p-value is small
Do not reject H0 because alpha is small
Reject H0 and conclude the population means differ
✓
Reject H0 and conclude the sample means are equal
Correct Answer
Reject H0 and conclude the population means differ
The p-value of 0.002 is below alpha = 0.01, so H0 is rejected and there is sufficient evidence that the two population means are different. Do not reject H0 because the p-value is small reverses the decision rule. Do not reject H0 because alpha is small ignores the comparison between the p-value and alpha. Reject H0 and conclude the sample means are equal contradicts itself, since rejecting H0 is evidence of a difference.
Category: Comparing Two Means
Question 20
In a matched pairs analysis, what is done with the two measurements in each pair?
Add them and analyze the sums
Analyze each group of values as a separate sample
Keep only the larger value from each pair
Subtract one from the other and analyze the differences
✓
Correct Answer
Subtract one from the other and analyze the differences
Each pair is reduced to one difference, computed in the same order for every pair, and the differences are then analyzed as a single sample with a t test. Add them and analyze the sums would not show how the two conditions compare. Analyze each group of values as a separate sample ignores the pairing and treats the data as independent. Keep only the larger value from each pair throws away the comparison the design was built to make.
Category: Matched Pairs