Question 1

The sample size in each independent sample must be the same if we are to test for differences between the means of two independent populations.

Accepted Answer

The sample sizes in each independent sample do not have to be the same to test for differences between the means of two independent populations. Statistical tests, such as the independent samples t-test, can accommodate different sample sizes, although equal sizes may increase the power of the test.

Question 2

The test for the equality of two population variances assumes that each of the two populations is normally distributed.

Accepted Answer

The test for the equality of two population variances is based on the F distribution, which assumes that the two populations are normally distributed. If this assumption is violated, the test may not be valid.

Question 3

When testing for the differences between the means of two independent populations,only a two-tailed test can be used.

Accepted Answer

Both a one-tailed test and a two-tailed test can be used when testing for the differences between the means of two independent populations, depending on the research question and hypothesis being tested.

Question 4

In testing for the differences between the means of two independent populations,we assume that the 2 populations each follow a normal distribution.

Accepted Answer

In testing for differences between the means of two independent populations, the assumption of normality is necessary in order to use parametric tests such as the t-test. Therefore, we assume that both populations are normally distributed.

Question 5

When the sample sizes are equal,the pooled variance of the two groups is the average of the two sample variances.

Accepted Answer

When the sample sizes are equal, the pooled variance is obtained by averaging the two sample variances. This is because the two groups are assumed to have equal variances under the null hypothesis.

Question 6

When testing H₀ : μ₁ - μ₂ = 0 against H₁ : μ₁ - μ₂ ≠ 0,the observed value of the Z-score was found to be -2.13.The p-value for this test would be 0.4668.

Accepted Answer

The p-value for a Z-score of -2.13 in a two-tailed test is approximately 0.0332, not 0.4668. This is because the Z-score indicates how many standard deviations an element is from the mean, and a Z-score of -2.13 corresponds to a p-value that reflects the probability of observing such a result under the null hypothesis. The p-value is found by looking up the Z-score in a standard normal distribution table or using a statistical software, and it represents the area under the curve to the extremes of the distribution. For a two-tailed test, you would look at both tails of the distribution.

Question 7

In testing H₀ : P_x - P_y = 0,an estimate of the common population proportion is a weighted average of the two sample proportions,the weights being equal to the sample sizes.

Accepted Answer

This statement is true. In order to estimate the common population proportion, we can calculate a pooled proportion by taking a weighted average of the two sample proportions, where the weights are equal to the sample sizes. This is known as the pooled proportion estimate.

Question 8

Repeated measurements collected from the same individuals is an example of data collected from two related populations.

Accepted Answer

When data is collected from the same individuals at different times or under different conditions, it is considered as data collected from two related populations. For example, measuring the blood pressure of the same individuals before and after a treatment. In this case, the data from two time points are related as they come from the same individuals.

Question 9

The F distribution is constructed as the ratio of two Student's t random variables;each divided by its degrees of freedom.

Accepted Answer

The F distribution is constructed as the ratio of two chi-squared distributions, each divided by its degrees of freedom, not Student's t random variables.

Question 10

When testing for differences between the means of two related populations,either a one-tailed or two-tailed test can be used.

Accepted Answer

This is true. Depending on the research question and hypothesis, either a one-tailed or two-tailed test can be used to test for differences between the means of two related populations. A one-tailed test is used when the research question predicts the direction of the difference (e.g. Population A will have a higher mean than Population B), while a two-tailed test is used when no specific direction is predicted.

Question 11

The F test used for testing the difference in two population variances is always a one-tailed test.

Accepted Answer

The F test used for testing the difference in two population variances is a two-tailed test where the null hypothesis is that the variances are equal and the alternative hypothesis is that the variances are not equal.

Question 12

At 5% significance level,the upper-tailed critical value of an F test with 3 degrees of freedom in the numerator and 8 degrees of freedom in the denominator is equal to 4.07.

Accepted Answer

This is true because when conducting an F test at 5% level of significance with 3 degrees of freedom in the numerator and 8 degrees of freedom in the denominator, the upper-tailed critical value is 4.07. Therefore, the statement is correct.

Question 13

An F distribution is an example of a symmetrical distribution.

Accepted Answer

An F distribution is not symmetrical, it is skewed to the right.

Question 14

To use the Student's t distribution to compare population means when the population variances are unknown and sample sizes are small,the populations are assumed to be normally distributed.

Accepted Answer

This is a correct statement. To use the Student's t distribution, it is assumed that the populations are normally distributed and that the population variances are unknown. Additionally, the sample sizes should be small (typically less than 30).

Question 15

In testing for differences between the means of two independent populations if the test statistic is very large,then the p-value of the test should be very small.

Accepted Answer

This is true because a large test statistic means that the difference between the sample means is large, which makes it less likely to have occurred by chance. Therefore, the p-value, which represents the probability of obtaining a test statistic as extreme or more extreme than the observed one under the null hypothesis, should be very small if the difference is statistically significant.

Question 16

In a random sample of 150 music industry management majors,75 rated a sense of humor as very important trait to their career performance.The same view was held by 81 of an independent random sample of 180 professional golf management majors.Test at the 5% level against a two-sided alternative the null hypothesis that the population proportions of music industry management and professional golf management majors who rate a sense of humor as very important are the same.

Accepted Answer

The answer of In a random sample of 150 music...

Question 17

In testing for the differences between the means of two related populations,we assume that the differences follow a t distribution.

Accepted Answer

The answer of In testing for the differences between the...

Question 18

In practical applications,the F ratio is arranged in a way such that the larger sample variance is in the denominator and the smaller is in the numerator.

Accepted Answer

In practical applications, the F ratio is arranged such that the larger sample variance is in the numerator and the smaller is in the denominator to ensure the ratio is greater than or equal to 1.

Question 19

The procedure that utilizes the normal distribution to test the difference between two population proportions is valid for all sample sizes.

Accepted Answer

The procedure that utilizes the normal distribution to test the difference between two population proportions requires that the sample sizes be large enough for the sampling distribution of the difference to be approximately normal. This is often assessed using the rule of thumb that both np and n(1-p) should be greater than 5 for both populations, where n is the sample size and p is the proportion.

Question 20

In testing H₀ : P_x - P_y = 0,the p-value equals the probability,1 - β,of avoiding the Type II error of erroneously failing to reject a null hypothesis that is in fact false.

Accepted Answer

The p-value measures the probability of observing a test statistic as extreme as, or more extreme than, the observed result under the assumption that the null hypothesis is true. It does not directly relate to the probability of avoiding a Type II error (1 - β), which is the power of the test.

Quiz 10: Hypothesis Testing: Additional Topics