Question 1

In constructing a confidence interval estimate for the difference between the means of two normally distributed populations, using two independent samples, we:

Accepted Answer

When the unknown population variances are equal, we can pool the sample variances to increase the sample size and improve the accuracy of our estimate. This results in a smaller standard error and a narrower confidence interval. However, if we have reason to believe that the population variances are unequal, we should not pool the sample variances and instead use a modified formula for the standard error.

Question 2

The quantity  is called the pooled variance estimate of the common variance of two unknown but equal population variances. It is the weighted average of the two sample variances, where the weights represent the:

Accepted Answer

The answer of The quantity  is called the pooled...

Question 3

Two samples of sizes 25 and 35 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:

Accepted Answer

The number of degrees of freedom for the equal-variances t-test statistic is calculated using the formula: df = (n1 + n2 - 2), where n1 and n2 are the sample sizes of the two groups. Thus, df = (25 + 35 - 2) = 58. Therefore, the best choice is C.

Question 4

Two independent samples are selected at random from two normal populations. The sample statistics are as follows:      and  Assume that a two-tailed hypothesis test is conducted at  , what is the value of the test statistic?

Accepted Answer

The answer of Two independent samples are selected at random...

Question 5

If we are testing for the difference between the means of two independent populations with equal variances, samples of  and  are taken, then the number of degrees of freedom is equal to:

Accepted Answer

To calculate the number of degrees of freedom for a two-sample t-test, we use the formula: 
df = n1 + n2 - 2 
where n1 and n2 are the sample sizes of the two populations. 
Plugging in the given values, we get: 
df = 11 + 11 - 2 = 20 
However, since we are assuming equal variances, we use a pooled estimate of the variance, which reduces the degrees of freedom by one. 
Thus, the correct answer is: 
df = 11 + 11 - 2 -1 = 19 
which is closest to option B).

Question 6

Which of the following is not a required assumption for constructing a confidence interval estimate of the difference between two population means when the samples are small?

Accepted Answer

The assumption that the population variances are equal to 1 is not required for constructing a confidence interval estimate of the difference between two population means when the samples are small. The required assumptions are A) the populations are normally distributed, B) the population variances are equal, and D) the samples are selected at random from the populations. Option E includes B and C, both of which are not required assumptions.

Question 7

In constructing 95% confidence interval estimate for the difference between the means of two normally distributed populations, where the unknown population variances are assumed not to be equal, summary statistics computed from two independent samples are as follows:  ,  ,  ,  ,  , and  . The upper confidence limit is:

Accepted Answer

The answer of In constructing 95% confidence interval estimate for...

Question 8

In testing the difference between the means of two normally distributed populations, the number of degrees of freedom associated with the unequal-variances t-test statistic usually results in a non-integer number. It is recommended that you:

Accepted Answer

When calculating the degrees of freedom for an unequal-variances t-test (also known as Welch's t-test), the resulting number often is a non-integer. The common practice is to round this number to the nearest integer to simplify the analysis, and rounding up is recommended to be more conservative in statistical testing.

Question 9

When we test for differences between the means of two independent populations, we can only use a two-tailed test.

Accepted Answer

We can use either a one-tailed or two-tailed test depending on the research question and hypothesis being tested.

Question 10

In testing the difference between two population means using two independent samples, the sampling distribution of the sample mean difference  is normal if:

Accepted Answer

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

Question 11

Two independent samples are selected at random from two normal populations. The sample statistics are as follows:      and  Assume that a two-tailed hypothesis test is conducted at  , what is the critical value?

Accepted Answer

The answer of Two independent samples are selected at random...

Question 12

Under what conditions can the t-distribution be correctly employed to test the difference between two population means?

Accepted Answer

The t-distribution can be correctly used under all the conditions listed: small, random samples (A), independence of samples (B), and equal or nearly equal population variances with normally distributed populations (C).

Question 13

In testing the difference between two population means, which of the following assumptions is not true for the two-sample procedure that uses a pooled estimate of the common variance  ?

Accepted Answer

The assumption that the populations from which the samples are drawn must be t-distributed is not true for the two-sample procedure that uses a pooled estimate of the common variance. Instead, the populations should be approximately normally distributed, and the sample sizes should be large (usually greater than 30) for the central limit theorem to work.

Question 14

Given the information:  ,  ,  ,  , the number of degrees of freedom that should be used in the pooled-variance t test is:

Accepted Answer

To determine the degrees of freedom for the pooled-variance t test, we need to add up the sample sizes from both groups (in this case, there are two groups: sample 1 and sample 2), and then subtract 2 to get the degrees of freedom.

Sample size (N1) for sample 1: 062d = 3, which means N1 = 3 hexadecimal = 3 × 16^0 = 3
Sample size (N2) for sample 2: 9f6b = 40043 hexadecimal = 262091 decimal
Total sample size: N1 + N2 = 3 + 262091 = 262094
Degrees of freedom: 262094 - 2 = 262092

Note: It is important to subtract 2 because we are estimating two unknown population parameters (the means) from the sample data.

Question 15

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:

Accepted Answer

When testing the difference between two population means with 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. This is because the pooled variance estimates the common variance of the two populations, allowing us to better estimate the standard error of the difference between means. If the populations are not normal or have unequal variances, we may need to use alternative methods such as the Welch's t-test. If the sample sizes are both large, we may be able to use the Central Limit Theorem to assume normality and estimate the standard error without pooling. However, this is not always reliable and it is generally safer to use the pooled variance when appropriate. Therefore, the best choice is B.

Question 16

Two independent samples of sizes 40 and 50 are randomly selected from two populations to test the difference between the population means  . The sampling distribution of the sample mean difference  is:

Accepted Answer

The answer of Two independent samples of sizes 40 and...

Question 17

The t-test for the difference between the means of two independent populations assumes that the respective:

Accepted Answer

The t-test for the difference between the means of two independent populations assumes that the populations from which the samples are drawn are approximately normally distributed. This assumption is crucial for the validity of the test results.

Question 18

In testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows:  ,  ,  ,  ,  , and  . Assume that the population variances are equal. Then, the standard error of the sampling distribution of the sample mean difference  is equal to:

Accepted Answer

The standard error of the sampling distribution of the sample mean difference can be calculated using the following formula:

SE = sqrt[(s1^2/n1) + (s2^2/n2)]

where s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes.

In this case, the summary statistics provided in the question only include the sample mean and sample size for each population, so we need to use the following formula to estimate the standard deviation of each population:

s = sqrt[((n-1) * S^2) / n]

where S is the sample standard deviation and n is the sample size.

Using this formula, we can estimate the standard deviation of each population as:

s1 = sqrt[((6-1) * 0.36) / 6] = 0.6
s2 = sqrt[((6-1) * 0.64) / 6] = 0.8

Now we can substitute these values into the formula for the standard error:

SE = sqrt[(s1^2/n1) + (s2^2/n2)] = sqrt[(0.6^2/6) + (0.8^2/6)] = 0.3189

Therefore, the best choice is C.

Question 19

Two independent samples of sizes 25 and 35 are randomly selected from two normal populations with equal variances. In order to test the difference between the population means, the test statistic is:

Accepted Answer

When comparing means from two independent samples (with equal variances) from normal populations, the test statistic follows a Student's t-distribution. The degrees of freedom (df) for this test are calculated using the formula df = n1 + n2 - 2, where n1 and n2 are the sample sizes. Here, df = 25 + 35 - 2 = 58.

Question 20

Two independent samples sizes 20 and 30 are randomly selected from two normally distributed populations. Assume that the population variances are unknown but equal. To test the difference between the population means,  , the sampling distribution of the sample mean difference,  , is:

Accepted Answer

The answer of Two independent samples sizes 20 and 30...

Quiz 10: Inference From Small Samples

In constructing a confidence interval estimate for the difference between the means of two normally distributed populations, using two independent samples, we:

The quantity is called the pooled variance estimate of the common variance of two unknown but equal population variances. It is the weighted average of the two sample variances, where the weights represent the:

Two samples of sizes 25 and 35 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:

Two independent samples are selected at random from two normal populations. The sample statistics are as follows: and Assume that a two-tailed hypothesis test is conducted at , what is the value of the test statistic?

If we are testing for the difference between the means of two independent populations with equal variances, samples of and are taken, then the number of degrees of freedom is equal to:

Which of the following is not a required assumption for constructing a confidence interval estimate of the difference between two population means when the samples are small?

In testing the difference between the means of two normally distributed populations, the number of degrees of freedom associated with the unequal-variances t-test statistic usually results in a non-integer number. It is recommended that you:

When we test for differences between the means of two independent populations, we can only use a two-tailed test.

In testing the difference between two population means using two independent samples, the sampling distribution of the sample mean difference is normal if:

Two independent samples are selected at random from two normal populations. The sample statistics are as follows: and Assume that a two-tailed hypothesis test is conducted at , what is the critical value?

Under what conditions can the t-distribution be correctly employed to test the difference between two population means?

In testing the difference between two population means, which of the following assumptions is not true for the two-sample procedure that uses a pooled estimate of the common variance ?

Given the information: , , , , the number of degrees of freedom that should be used in the pooled-variance t test is:

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:

Two independent samples of sizes 40 and 50 are randomly selected from two populations to test the difference between the population means . The sampling distribution of the sample mean difference is:

The t-test for the difference between the means of two independent populations assumes that the respective:

In testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows: , , , , , and . Assume that the population variances are equal. Then, the standard error of the sampling distribution of the sample mean difference is equal to:

Two independent samples of sizes 25 and 35 are randomly selected from two normal populations with equal variances. In order to test the difference between the population means, the test statistic is:

Two independent samples sizes 20 and 30 are randomly selected from two normally distributed populations. Assume that the population variances are unknown but equal. To test the difference between the population means, , the sampling distribution of the sample mean difference, , is: