Question 1

If the limits of the confidence interval of the difference between the means of two normally distributed populations were from -2.6 to 1.4 at the 95 percent confidence level,then we can conclude that we are 95 percent certain that there is a significant difference between the two population means.

Accepted Answer

The confidence interval includes zero (0), which means we cannot confidently conclude that there is a significant difference between the two population means. A confidence interval that does not include zero would suggest that there is a significant difference between the two population means.

Question 2

In forming a confidence interval for μ₁ - μ₂,only two assumptions are required: independent samples and sample sizes of at least 30.

Accepted Answer

The assumptions are independent samples, equal variances, and either sample sizes are large or the sampled populations are normally distributed.

Question 3

In testing the difference between the means of two normally distributed populations using large independent random samples,the sample sizes from the two populations must be equal.

Accepted Answer

Sample sizes do not have to be equal to test the differences of the means.

Question 4

In testing the difference between two means from two independent populations,the sample sizes do not have to be equal.

Accepted Answer

This is true. T-tests can be conducted with unequal sample sizes as long as the assumptions of normality, independence, and equal variances (for a two-sample t-test) are met. There are also modified t-tests that can be used when these assumptions are not met.

Question 5

In testing the difference between the means of two normally distributed populations using independent random samples,the alternative hypothesis always indicates no differences between the two specified means.

Accepted Answer

The alternative hypothesis in a two-sample t-test indicates that there is a significant difference between the means of the two populations being compared. It does not indicate that there is no difference.

Question 6

Assume that we are constructing a confidence interval for the difference in the means of two populations based on independent random samples.If both sample sizes n₁ and n₂ = 10,and the distributions of both populations are highly skewed,then a confidence interval for the difference in the means can be constructed using the t test statistic.

Accepted Answer

If the distributions of both populations are highly skewed and the sample sizes are small, then the t test may not be appropriate. In such cases, a nonparametric test like the Wilcoxon rank-sum test may be more appropriate.

Question 7

An independent samples experiment is an experiment in which there is no relationship between the measurements in the different samples.

Accepted Answer

In an independent samples experiment, the samples are chosen in such a way that they are unrelated or independent of each other. Therefore, the measurements in the different samples do not affect each other.

Question 8

When we are testing a hypothesis about the difference in two population proportions based on large independent samples,we compute a combined (pooled)proportion from the two samples if we assume that there is no difference between the two proportions in our null hypothesis.

Accepted Answer

This statement is true. A pooled proportion is calculated when assuming that there is no difference between two population proportions in the null hypothesis. It is used to estimate the common proportion of success for both populations.

Question 9

In testing the difference between the means of two independent populations,if neither population is normally distributed,then the sampling distribution of the difference in means will be approximately normal,provided that the sum of the sample sizes obtained from the two populations is at least 30.

Accepted Answer

We need each sample to be large (at least greater than 30).

Question 10

When comparing the variances of two normally distributed populations using independent random samples,if s₁² = s₂²,the calculated value of F will always be equal to one.

Accepted Answer

If the variances of two normally distributed populations are equal and we take independent random samples from each of them, the calculated value of F will always be equal to one. This is known as the null hypothesis in the F-test. Therefore, the statement is true.

Question 11

In testing the equality of population variances,two assumptions are required: independent samples and normally distributed populations.

Accepted Answer

Both assumptions of independent samples and normally distributed populations are required for testing the equality of population variances using methods such as the F-test or Levene's test.

Question 12

The controller of a chain of toy stores is interested in determining whether there is any difference in the weekly sales of store 1 and store 2.The weekly sales are normally distributed.This problem should be analyzed using an independent means method.

Accepted Answer

An independent means method, also known as independent samples t-test, is used when comparing the means of two independent groups. In this case, the weekly sales of store 1 and store 2 are two independent groups, which can be compared using the independent means method. Therefore, the correct answer is A - True.

Question 13

When comparing two population means based on independent random samples,the pooled estimate of the variance is used when there is an assumption of equal population variances.

Accepted Answer

This statement is true. The pooled estimate of the variance is used when the assumption of equal population variances is made in order to combine the variances from the two samples to get a more accurate estimate of the variance of the population means.

Question 14

In testing the difference between two population variances,it is a common practice to compute the F statistic so that its value is always greater than or equal to one.

Accepted Answer

This is true because the F statistic is the ratio of two variances, and by definition, variances are always non-negative. Therefore, the numerator (which is the sample variance of one population) divided by the denominator (which is the sample variance of the other population) is always greater than or equal to one.

Question 15

In an experiment involving matched pairs,a sample of 12 pairs of observations is collected.The degrees of freedom for the t statistic is 10.

Accepted Answer

Degrees of freedom is n − 1 = 12 − 1 = 11

Question 16

When testing the difference between two proportions selected from populations with large independent samples,the z test statistic is used.

Accepted Answer

This statement is true. When comparing two proportions from large independent samples, the z-test statistic is appropriate because the sampling distribution of the difference between two proportions is approximately normal.

Question 17

When comparing two independent population means,if n₁ = 13 and n₂ = 10,degrees of freedom for the t statistic is 22.

Accepted Answer

The formula for degrees of freedom for independent two-sample t-test is df = (n1-1) + (n2-1), where n1 and n2 are sample sizes of population 1 and 2 respectively. Using the given sample sizes, we can calculate df as (13-1) + (10-1) = 21, not 22. Therefore, the statement is false.

Question 18

The F statistic can assume either a positive or a negative value.

Accepted Answer

The F statistic is always non-negative.

Question 19

In testing the difference between the means of two normally distributed populations using independent random samples,we can only use a two-sided test.

Accepted Answer

We can use a one-sided test if we have a prior hypothesis about the direction of the difference in means. However, if we have no prior hypothesis, we use a two-sided test.

Question 20

If the limits of the confidence interval of the difference between the means of two normally distributed populations were 8.5 and 11.5 at the 95 percent confidence level,then we can conclude that we are 95 percent certain that there is a significant difference between the two population means.

Accepted Answer

The confidence interval of the difference between the means does not include zero, which means that the difference between the two population means is statistically significant at the 95 percent confidence level. Therefore, we can conclude with 95 percent confidence that there is a significant difference between the two population means.

Quiz 10: Statistical Inferences Based on Two Samples