Question 1

In order to construct an interval estimate for the difference in the means of two normally distributed populations with unknown but equal variances, using two independent samples of size n₁ and n₂, we must use a t distribution with (n₁+ n₂)degrees of freedom.

Accepted Answer

When the variances are unknown but assumed equal, the t distribution used for the interval estimate has (n₁ + n₂ - 2) degrees of freedom, not (n₁ + n₂).

Question 2

If a 90% confidence interval for the difference in the two population means contains zero, then the null hypothesis of zero difference between the two population means cannot be rejected at a 0.10 level of significance.

Accepted Answer

This statement is true. A 90% confidence interval for the difference in the two population means that contains zero indicates that there is a 90% chance that the true difference in population means falls within that range. If the range includes zero, it suggests that there may not be a significant difference between the two populations, and the null hypothesis cannot be rejected at a significance level of 0.10 or higher. A 90% confidence interval containing zero means that the difference between the two population means is not statistically significant at a 0.10 level of significance, so the null hypothesis cannot be rejected.

Question 3

The F test of two population variances is extremely robust to the violations of the assumption that the populations are normally distributed.

Accepted Answer

The F test of two population variances is sensitive to violations of the assumption of normality, meaning its results can be unreliable if the populations are not normally distributed.

Question 4

If a 98% confidence interval for the difference in the two population means does not contain zero, then the null hypothesis of zero difference between the two population means cannot be rejected at a 0.02 level of significance.

Accepted Answer

If a 98% confidence interval for the difference in the two population means does not contain zero, then the null hypothesis of zero difference between the two population means can be rejected at a significance level of 0.02. This is because a 98% confidence interval corresponds to a significance level of 0.02 for a two-tailed test. If a 98% confidence interval for the difference in the two population means does not contain zero, it means that the difference between the two population means is statistically significant at a 0.02 level of significance. Therefore, the null hypothesis of zero difference between the two population means can be rejected at a 0.02 level of significance. The correct answer, therefore, is B.

Question 5

If the variances of the two populations are not equal, it is appropriate to use the "unpooled" formula to determine the t statistic for the hypothesis test of the difference in the two population means.

Accepted Answer

When the variances of the two populations are not equal, it is appropriate to use the "unpooled" formula for calculating the standard error of the difference between the means. This involves using separate estimates of the population variances for each group, rather than assuming they are equal and pooling them together. This results in a modified t statistic that takes into account the differences in variances between the two groups.

Question 6

Sets of matched samples are also referred to as independent samples.

Accepted Answer

Sets of matched samples are actually referred to as dependent samples, not independent samples. Independent samples are samples that are unrelated to each other and are selected randomly without any relationship or dependency.

Question 7

If the populations are normally distributed but the population variances are unknown, the t statistic can be used as the basis for statistical inferences about the difference in two population means using two independent random samples.

Accepted Answer

This statement is true. When the population variances are unknown, the t statistic is used instead of the z statistic to test hypotheses and make inferences about the difference in two population means using two independent random samples. This statement is true. When the population variances are not known, the t-test is a common method for comparing the means of two independent samples.

Question 8

Restaurateur Denny Valentine is evaluating two sites, Raymondville and Rosenberg, for his next restaurant.He wants to prove that Raymondville residents (population 1)dine out more often than Rosenberg residents (population 2).Denny plans to test this hypothesis using a random sample of 81 families from each suburb.His null hypothesis is ___. A)$\sigma$₁² < $\sigma$₂² B)$\mu$₁ - µ₂> 0 C)p₁ - p₂ = 0 D)$\mu$₁ - $\mu$₂ = 0 E)s₁ - s₂ = 0

Accepted Answer

The null hypothesis typically states that there is no difference between the groups being compared. In this case, it would be that the mean dining out frequency of Raymondville residents is equal to that of Rosenberg residents, which is represented by $\mu$₁ - $\mu$₂ = 0.

Question 9

In a set of matched samples, each data value in one sample is related to or matched with a corresponding data value in the other sample.

Accepted Answer

In a set of matched samples, each data value in one sample is related to or matched with a corresponding data value in the other sample. Therefore, the statement is true. In matched samples, each data value in one sample is related to or paired with a corresponding data value in the other sample, making the samples directly comparable.

Question 10

If the sample sizes are small, but the populations are normally distributed and the population variances are known, the z statistic can be used as the basis for statistical inferences about the difference in two population means using two independent random samples.

Accepted Answer

When the population variances are known and the sample sizes are small, the z statistic can be used to make inferences about the difference in two population means using two independent random samples. This assumption is valid when the populations are normally distributed. When sample sizes are small and populations are normally distributed with known variances, the z statistic can be used to make inferences about the difference in population means using independent random samples.

Question 11

The difference in two sample means is normally distributed for sample sizes ≥ 30, only if the populations are normally distributed.

Accepted Answer

According to the Central Limit Theorem, the distribution of the sample means will approximate a normal distribution as the sample size becomes large (usually n ≥ 30) regardless of the shape of the population distribution.

Question 12

Restaurateur Denny Valentine is evaluating two sites, Raymondville and Rosenberg, for his next restaurant.He wants to prove that Raymondville residents (population 1)dine out more often than Rosenberg residents (population 2).Denny plans to test this hypothesis using a random sample of 81 families from each suburb.His alternate hypothesis is ___. A)$\sigma$₁² < $\sigma$₂² B)$\mu$₁- $\mu$₂> 0 C)p₁ - p₂ = 0 D)$\mu$₁ - $\mu$₂ = 0 E)s₁ - s₂ > 0

Accepted Answer

The alternate hypothesis is that the mean number of times that Raymondville residents dine out is greater than the mean number of times that Rosenberg residents dine out.

Question 13

In order to construct an interval estimate for the difference in the means of two normally distributed populations with unknown but equal variances, using two independent samples of size n₁ and n₂, we must use a t distribution with (n₁+ n₂− 2)degrees of freedom.

Accepted Answer

This statement is true. When constructing an interval estimate for the difference in means of two normally distributed populations with unknown but equal variances, we use a t distribution with (n1 + n2 - 2) degrees of freedom.

Question 14

If the populations are normally distributed but the population variances are unknown, the z statistic can be used as the basis for statistical inferences about the difference in two population means using two independent random samples.

Accepted Answer

When the population variances are unknown and the populations are normally distributed, the t-statistic, not the z-statistic, is used for making inferences about the difference between two population means using two independent samples.

Question 15

If the variances of the two populations are not equal, it is appropriate to use the "pooled" formula to determine the t statistic for the hypothesis test of the difference in the two population means.

Accepted Answer

The "pooled" formula for the t-statistic assumes that the variances of the two populations are equal. When the variances are not equal, an alternative approach, such as Welch's t-test, should be used.

Question 16

Sets of matched samples are also referred to as dependent samples.

Accepted Answer

Sets of matched samples are dependent on each other, hence they are also known as dependent samples. Sets of matched samples are dependent because they are related to each other through the matching process. The values of one sample are linked or connected to the values of the other sample, which makes them dependent.

Question 17

To determine if there is a difference in the strength of steel produced from two different production processes, a process manager will draw independent samples from the two processes and compare the difference in the sample means.

Accepted Answer

The described process is a valid method for testing for a difference in the strength of steel produced from two different production processes.

Question 18

To test hypotheses about the equality of two population variances, the ratio of the variances of the samples from the two populations is tested using the F test.

Accepted Answer

The F test is used to test hypotheses about the equality of two population variances by comparing the ratio of the variances of the samples from the two populations. Therefore, the statement is true. This statement is true, the F test is used to test hypotheses about the equality of two population variances by comparing the ratio of the variances of the samples from the two populations.

Question 19

If the variances of the two populations are equal, it is appropriate to use the "pooled" formula to determine the t statistic for the hypothesis test of the difference in the two population means.

Accepted Answer

When the variances of the two populations are equal, it is appropriate to use the pooled formula to determine the t statistic for the hypothesis test of the difference in the two population means. This involves combining the sample variances and degrees of freedom using a weighted average to calculate the pooled variance, which is then used to calculate the standard error of the difference between means. This pooled standard error is used in the t statistic formula, which assumes equal variances in the two populations.

Question 20

If the sample sizes are greater than 30 and the population variances are known, the basis for statistical inferences about the difference in two population means using two independent random samples is the z statistic, regardless of the shapes of the population distributions.

Accepted Answer

When the sample sizes are greater than 30 and the population variances are known, the z statistic can be used to make statistical inferences about the difference in two population means, regardless of the shapes of the population distributions. When sample sizes are greater than 30 and population variances are known, the sampling distribution of the sample means is assumed to be normally distributed due to the central limit theorem. Therefore, the z statistic can be used to make inferences about the difference in two population means. This is true regardless of the shape of the population distributions.

Quiz 10: Statistical Inferences About Two Populations

In order to construct an interval estimate for the difference in the means of two normally distributed populations with unknown but equal variances, using two independent samples of size n1 and n2, we must use a t distribution with (n1 + n2)degrees of freedom.