Question 1

Ten functionally illiterate adults were given an experimental one-week crash course in reading. Each of the ten adults was given a reading test prior to the course and another test after the course. The results are shown below.  Is there enough evidence to infer at the 5% significance level that the reading scores have improved?
Use the Data Analysis software if you prefer.
Test statistic = ______________
Critical Value(s) = ______________
Conclusion: ______________
Interpretation: __________________________________________
Estimate the mean improvement with 95% confidence.
______________

Accepted Answer

The answer of Ten functionally illiterate adults were given an...

Question 2

Statisticians have shown that for given sample sizes  and  , the number of degrees of freedom associated with the equal-variances test statistic and confidence interval estimator of  is always greater than or equal to number of degrees of freedom associated with the unequal-variances test statistic and confidence interval estimator.

Accepted Answer

This is a well-established fact in statistics. When sample sizes are equal, the variance estimates for the two samples are pooled, resulting in a larger degrees of freedom for the equal-variances test statistic and confidence interval estimator compared to the unequal-variances test statistic and confidence interval estimator.

Question 3

A political analyst in Iowa surveys a random sample of registered Democrats and compares the results with those obtained from a random sample of registered Republicans. This would be an example of two independent samples.

Accepted Answer

The two groups being compared (registered Democrats vs. registered Republicans) are independent of each other, and therefore this is an example of two independent samples.

Question 4

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population means, but your statistical software provides only a one-tail area of 0.045 as part of its output. The p-value for this test will be 0.09.

Accepted Answer

In a two-tail test, the p-value is calculated by doubling the one-tail area, so if the one-tail area is 0.045, the two-tail p-value would indeed be 0.09.

Question 5

Both the equal-variances and unequal variances test statistic and confidence interval estimator of  require that the two populations be normally distributed.

Accepted Answer

Both the equal-variances and unequal variances test statistic and confidence interval estimator assume that the two populations being compared are normally distributed. This assumption is often checked through normal probability plots or other diagnostic tests.

Question 6

The mean value of  is equal to  .

Accepted Answer

If the mean value of two values is equal, then the two values themselves must be equal. Therefore, 11eb6b9e_062a_43d5_8d9d_c98c51ab307f_TB8220_11 = 11eb6b9e_062a_43d6_8d9d_af376fce74ac_TB8220_11, and so their mean value will also be equal.

Question 7

The general manager of a chain of fast food chicken restaurants wants to determine how effective their promotional campaigns are. In these campaigns "20% off" coupons are widely distributed. These coupons are only valid for one week. To examine their effectiveness, the executive records the daily gross sales (in $1,000s) in one restaurant during the campaign and during the week after the campaign ends. The data is shown below.  Can they infer at the 5% significance level that sales increase during the campaign?
Use the Data Analysis software if you prefer.
Test statistic = ______________
Critical Value(s) = ______________
Conclusion: ______________
Interpretation: __________________________________________
Estimate with 95% confidence the mean difference.
______________

Accepted Answer

The answer of The general manager of a chain of...

Question 8

The pooled-variances t-test requires that the two population variances need not be the same.

Accepted Answer

The pooled-variances t-test assumes that the two population variances are equal, which is a requirement for pooling the variances.

Question 9

The variance of  is equal to  .

Accepted Answer

The answer of The variance of  is equal to...

Question 10

Two samples of sizes 15 and 20 are randomly and independently selected from two normally distributed populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom associated with the two-sample t-test is equal to 35.

Accepted Answer

The correct number of degrees of freedom for the two-sample t-test with equal variances is (n1 + n2 - 2), which in this case would be (15 + 20 - 2) = 33.

Question 11

In testing the difference between two population means using two independent samples, the sampling distribution of the sample mean difference  is normal if the sample sizes are both greater than 30.

Accepted Answer

The answer of In testing the difference between two population...

Question 12

Automobile insurance appraisers examine cars that have been involved in accidental collisions to assess the cost of repairs. An insurance executive is concerned that different appraisers produce significantly different assessments. In an experiment 10 cars that have recently been involved in accidents were shown to two appraisers. Each assessed the estimated repair costs. These results are shown below.  Can the executive conclude at the 5% significance level that the appraisers differ in their assessments?
Use the Data Analysis software if you prefer.
Test statistic = ______________
Critical Value(s) = ______________
Conclusion: ______________
Interpretation: __________________________________________

Accepted Answer

The answer of Automobile insurance appraisers examine cars that have...

Question 13

Two samples of sizes 25 and 20 are independently drawn from two normal populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t-test statistic is 44.

Accepted Answer

The degrees of freedom for the equal-variances t-test is calculated as the sum of the sample sizes minus 2, which is (25 + 20 - 2) = 43.

Question 14

In testing the difference between two population means using two independent samples, the population standard deviations are assumed to be known, and the calculated test statistic equals 2.75. If the test is two-tail and 5% level of significance has been specified, the conclusion should not be reject the null hypothesis.

Accepted Answer

Given the test statistic of 2.75 and a two-tailed test at a 5% level of significance, we typically compare the test statistic to critical values from the standard normal distribution (Z-distribution) because the population standard deviations are assumed to be known. For a 5% level of significance in a two-tailed test, the critical values are approximately ±1.96. Since 2.75 is greater than 1.96, the null hypothesis should be rejected.

Question 15

When the population variances are unequal, we estimate each population variance with its sample variance. Hence, the unequal-variances test statistic of  is Student t distributed with  +  -2 degrees of freedom.

Accepted Answer

The answer of When the population variances are unequal, we...

Question 16

A marketing consultant was in the process of studying the perceptions of married couples concerning their weekly food expenditures. He believed that the husband's perception would be higher than the wife's. To judge his belief, he takes a random sample of ten married couples and asks each spouse to estimate the family food expenditure (in dollars) during the previous week. The data are shown below.  Can the consultant conclude at the 5% significance level that the husband's estimate is higher than the wife's estimate?
Use the Data Analysis software if you prefer.
Test statistic = ______________
Critical Value(s) = ______________
Conclusion: ______________
Interpretation: __________________________________________
Estimate with 95% confidence the population mean difference.
______________

Accepted Answer

The answer of A marketing consultant was in the process...

Question 17

In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference  if the populations are normal with equal variances.

Accepted Answer

When testing the difference between two population means using two independent samples and assuming normality and equal variances, we use the pooled variance to estimate the standard error of the sampling distribution of the sample mean difference. This is because the pooled variance is a weighted average of the sample variances from each group, which accounts for the variation within each group and provides a more accurate estimate of the population variance.

Question 18

The equal-variances test statistic of  is Student t distributed with  +  degrees of freedom, provided that the two populations are normal.

Accepted Answer

The answer of The equal-variances test statistic of  is...

Question 19

The sample size in each independent sample must be the same if we are to test for differences between the means of 2 independent populations.

Accepted Answer

While having equal sample sizes can increase the power of a statistical test, it is not a requirement for testing differences between the means of two independent populations. Tests like the independent samples t-test can be applied even when sample sizes differ.

Question 20

The sampling distribution of  is normal if the sampled populations are normal, and approximately normal if the populations are nonnormal and the sample sizes  and  are large.

Accepted Answer

This statement is true according to the central limit theorem, which states that for large sample sizes, the sampling distribution of means will be approximately normal regardless of the underlying distribution of the population. For smaller sample sizes, the distribution of the population does have an effect on the sampling distribution.

Quiz 10: Inference From Small Samples

A political analyst in Iowa surveys a random sample of registered Democrats and compares the results with those obtained from a random sample of registered Republicans. This would be an example of two independent samples.

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population means, but your statistical software provides only a one-tail area of 0.045 as part of its output. The p-value for this test will be 0.09.

Both the equal-variances and unequal variances test statistic and confidence interval estimator of require that the two populations be normally distributed.

The mean value of is equal to .

The pooled-variances t-test requires that the two population variances need not be the same.

The variance of is equal to .

Two samples of sizes 15 and 20 are randomly and independently selected from two normally distributed populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom associated with the two-sample t-test is equal to 35.

In testing the difference between two population means using two independent samples, the sampling distribution of the sample mean difference is normal if the sample sizes are both greater than 30.

Two samples of sizes 25 and 20 are independently drawn from two normal populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t-test statistic is 44.

When the population variances are unequal, we estimate each population variance with its sample variance. Hence, the unequal-variances test statistic of is Student t distributed with + -2 degrees of freedom.

In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference if the populations are normal with equal variances.

The equal-variances test statistic of is Student t distributed with + degrees of freedom, provided that the two populations are normal.

The sample size in each independent sample must be the same if we are to test for differences between the means of 2 independent populations.

The sampling distribution of is normal if the sampled populations are normal, and approximately normal if the populations are nonnormal and the sample sizes and are large.