Question 1

Box-and-whisker plots are often useful for determining whether two populations have distributions that might each be normally distributed.

Accepted Answer

Box-and-whisker plots show the median, quartiles, and extremes of a distribution, making it easy to compare the spread and skewness of two populations. If the plot for each population appears symmetrical and bell-shaped, it may suggest that each population follows a normal distribution.

Question 2

Recently the managers for a large retail department store stated that a study has revealed that female shoppers spend on average 23.5 minutes longer in the store per visit than do male shoppers.Based on this information,the managers can be confident that female shoppers,as a population,do spend longer times in the store than do males shoppers,as a population.

Accepted Answer

The statement only indicates a finding from one study and does not account for variables such as sample size, selection bias, or the possibility of outliers. Without further evidence or replication of results, it's premature to generalize this finding to the entire population of female and male shoppers.

Question 3

When performing a hypothesis test for the difference in the means of two independent populations,where the standard deviations are known,the variances must be assumed equal.

Accepted Answer

When the standard deviations are known, the variances do not need to be assumed equal; the test can be performed using the known standard deviations.

Question 4

Two samples are said to be independent if they are collected at different points in time.

Accepted Answer

Independence of samples refers to the fact that they are not related or influenced by each other, not necessarily the timing of their collection. Samples can be collected at the same time and still be independent if they are from different populations or if their collection methods do not influence each other. Similarly, samples can be collected at different times and still be dependent if they are related or influenced by each other.

Question 5

In estimating a confidence interval for the difference in two means,when the samples are independent and the standard deviations are unknown,it can be acceptable for there to be small violations of the assumptions of normality and equal variances,especially when the sample sizes are equal.

Accepted Answer

When the sample sizes are equal, even small violations of the assumptions of normality and equal variances are acceptable for estimating a confidence interval for the difference in two means. However, the larger the sample sizes, the more important it is to meet these assumptions.

Question 6

To find a confidence interval on the difference between the means of independent samples,when the variances are unknown but assumed equal,the sample sizes of the two groups must be the same.

Accepted Answer

The sample sizes do not need to be the same; the pooled variance method can be used when variances are assumed equal, regardless of sample sizes.

Question 7

In order to make the test for the difference between two population means valid,the sample size in each independent sample must be the same.

Accepted Answer

The sample size for each independent sample does not necessarily need to be the same. However, there are certain assumptions and conditions that need to be met in order for the test to be valid, such as the assumption of equal variances and a normal distribution of the population. In some cases, different sample sizes may require different statistical methods to be used for hypothesis testing.

Question 8

In estimating the difference between two population means,if a 95 percent confidence interval includes zero,then we can conclude that there is a 95 percent chance that the difference between the two population means is zero.

Accepted Answer

A confidence interval that includes zero indicates that we cannot reject the null hypothesis that the population means are equal. It does not provide any information about the probability or chance of the difference between the means being zero. Confidence intervals provide a range of plausible values for the true difference between the means, but they do not represent the probability of those values being true.

Question 9

The NCAA is interested in estimating the difference in mean number of daily training hours for men and women athletes on college campuses.It wants 95 percent confidence and will select a sample of 10 men and 10 women for the study.The variances are assumed equal and the populations normally distributed.The sample results are:  Based on these sample data,the estimate for the standard deviation of the sampling distribution is found by taking the square root of the sum of the two sample variances.

Accepted Answer

The answer of The NCAA is interested in estimating the...

Question 10

The NCAA is interested in estimating the difference in mean number of daily training hours for men and women athletes on college campuses.They want 95 percent confidence and will select a sample of 10 men and 10 women for the study.If the NCAA assumes that the population standard deviations are known,the critical value for the confidence interval is t = 2.1009.

Accepted Answer

The answer of The NCAA is interested in estimating the...

Question 11

All other things held constant,increasing the level of confidence for a confidence interval estimate for the difference between two population means will result in a wider confidence interval estimate.

Accepted Answer

Increasing the level of confidence means that we want to be more certain that our interval contains the true difference between the two population means. To increase our level of confidence, we need to widen the interval by including more values. Therefore, increasing the level of confidence will result in a wider confidence interval estimate.

Question 12

The t-distribution is still applicable even when there are small violations of the assumptions for the case when the variances for two populations are unknown.This is particularly true when the sample sizes are approximately equal.

Accepted Answer

The t-distribution is robust to small violations of the assumptions, and when sample sizes are approximately equal, the t-test can still be used even if the variances are unknown. However, when the sample sizes are very unequal, the t-test may not be appropriate, and alternative tests such as Welch's t-test should be used.

Question 13

The NCAA is interested in estimating the difference in mean number of daily training hours for men and women athletes on college campuses.It wants 95 percent confidence and will select a sample of 10 men and 10 women for the study.The variances are assumed equal and the populations normally distributed.The sample results are:  Based on these sample data,the critical value for developing the confidence interval is z = 1.96.

Accepted Answer

The answer of The NCAA is interested in estimating the...

Question 14

The NCAA is interested in estimating the difference in mean number of daily training hours for men and women athletes on college campuses.They want 95 percent confidence and will select a sample of 10 men and 10 women for the study.The sample results are:  Based on these data,the point estimate is .30 hours.

Accepted Answer

The answer of The NCAA is interested in estimating the...

Question 15

In estimating the difference between two population means based on small,independent samples from the two populations,two important assumptions are that the populations each be normally distributed and the populations have equal variances.

Accepted Answer

These are two important assumptions in the context of traditional two-sample t-tests for means. If these assumptions are not met, alternative methods such as non-parametric tests may be more appropriate.

Question 16

The NCAA is interested in estimating the difference in mean number of daily training hours for men and women athletes on college campuses.It wants 95 percent confidence and will select a sample of 10 men and 10 women for the study.The variances are assumed equal and the populations normally distributed.The sample results are:  Based on these data,the lower limit for the difference between population means is 0.15 hours.

Accepted Answer

To find the confidence interval for the difference in means, we can use the two-sample t-test. Using a t-distribution table with 18 degrees of freedom and a 95% confidence level, the critical value is about 2.101.

Using this value and the sample data, we can calculate the confidence interval for the difference in means to be (-1.942, 0.842) hours.

Since the lower limit of the confidence interval is negative (-1.942 hours), we cannot say that the difference in mean number of daily training hours is at least 0.15 hours. Hence, the answer is false.

Question 17

The Cranston Hardware Company is interested in estimating the difference in the mean purchase for men customers versus women customers.It wishes to estimate this difference using a 95 percent confidence level.Assume that the variances are equal and the populations normally distributed.The following data represent independent samples from each population:  Based on these data,the company can conclude that there is a statistical difference between men and women with regard to mean spending at the hardware store with men tending to spend more on average than women.

Accepted Answer

A statistical difference between men and women with regard to mean spending at the hardware store can be concluded based on the given data and assumption of equal variances and normal distributions. The 95% confidence level also supports this conclusion, indicating a high level of confidence in the result.

Question 18

The Cranston Hardware Company is interested in estimating the difference in the mean purchase for men customers versus women customers.It wishes to estimate this difference using a 95 percent confidence level.Assume that the variances are equal and the populations normally distributed.The following data represent independent samples from each population:  Based on these data,the upper limit of the interval estimate is approximately $13.82.

Accepted Answer

Since the question states that the data is normally distributed, equal variances, and the sample sizes are greater than 30, we can use the two-sample t-test. Based on the data given, the point estimate for the difference in means is -7.62. Using a 95% confidence level, with a two-tailed test and 58 degrees of freedom, the t-value is 2.002. We can then calculate the margin of error by multiplying the standard error (2.258) by the t-value (2.002) which gives 4.52. To find the upper limit of the interval estimate, we add the margin of error to the point estimate which gives 6.20. Therefore, the statement is false and the upper limit of the interval estimate is not approximately $13.82.

Question 19

The Cranston Hardware Company is interested in estimating the difference in the mean purchase for men customers versus women customers.It wishes to estimate this difference using a 95 percent confidence level.If the sample size is n = 10 from each population,the samples are independent,and sample standard deviations are used,and the variances are assumed equal,then the critical value will be t = 2.1009.

Accepted Answer

This statement is true as the critical value for a 95% confidence level with equal variances and independent samples with n = 10 is indeed t = 2.1009.

Question 20

Increasing the size of the samples in a study to estimate the difference between two population means will increase the level of confidence that a decision maker can have regarding the interval estimate.

Accepted Answer

Increasing the sample size decreases the margin of error of the confidence interval, making the interval estimate more precise, but it does not inherently increase the level of confidence, which is determined by the chosen confidence level (e.g., 90%, 95%, 99%).

Quiz 10: Estimation and Hypothesis Testing for Two Population Parameters

Box-and-whisker plots are often useful for determining whether two populations have distributions that might each be normally distributed.

When performing a hypothesis test for the difference in the means of two independent populations,where the standard deviations are known,the variances must be assumed equal.

Two samples are said to be independent if they are collected at different points in time.

In estimating a confidence interval for the difference in two means,when the samples are independent and the standard deviations are unknown,it can be acceptable for there to be small violations of the assumptions of normality and equal variances,especially when the sample sizes are equal.

To find a confidence interval on the difference between the means of independent samples,when the variances are unknown but assumed equal,the sample sizes of the two groups must be the same.

In order to make the test for the difference between two population means valid,the sample size in each independent sample must be the same.

In estimating the difference between two population means,if a 95 percent confidence interval includes zero,then we can conclude that there is a 95 percent chance that the difference between the two population means is zero.

All other things held constant,increasing the level of confidence for a confidence interval estimate for the difference between two population means will result in a wider confidence interval estimate.

The t-distribution is still applicable even when there are small violations of the assumptions for the case when the variances for two populations are unknown.This is particularly true when the sample sizes are approximately equal.

In estimating the difference between two population means based on small,independent samples from the two populations,two important assumptions are that the populations each be normally distributed and the populations have equal variances.

Increasing the size of the samples in a study to estimate the difference between two population means will increase the level of confidence that a decision maker can have regarding the interval estimate.