Question 1

The pooled-variance estimator, $s _ { p } ^ { 2 }$ , requires that the two population variances be equal.

Accepted Answer

The pooled-variance estimator assumes equal population variances to combine sample variances into a single estimate.

Question 2

A one-tail test of the population proportion produces a test statistic z = -2.12. The p-value of the test is 0.034.

Accepted Answer

A one-tail test with a z-score of -2.12 would have a p-value of approximately 0.017, not 0.034, as the p-value represents the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis.

Question 3

In testing the hypotheses: H₀ : $\mu$ = 140 H₁ : $\mu$ $\neq$140, Suppose that we rejected the null hypothesis at $\alpha$ = 0.10. Then for which of the following$\alpha$ values do we also reject the null hypothesis?

Accepted Answer

If we reject the null hypothesis at α = 0.10, then we would also reject the null hypothesis at any smaller level of significance. So, all of the given choices are correct.

Question 4

A sample of size 125 selected from one population has 55 successes, and a sample of size 140 selected from a second population has 70 successes. The test statistic for testing the equality of the population proportions is equal to:

Accepted Answer

The formula for the test statistic for testing the equality of the population proportions is:

z = (p1 - p2) / sqrt [ (p̂(1-p̂) / n1) + (p̂(1-p̂) / n2) ]

where p1 and p2 are the population proportions, p̂ is the pooled sample proportion, and n1 and n2 are the sample sizes.

Plugging in the given values, we get:

p̂ = (x1 + x2) / (n1 + n2) = (55 + 70) / (125 + 140) = 0.514

z = (0.55 - 0.5) / sqrt [ (0.514(1-0.514) / 125) + (0.514(1-0.514) / 140) ]
  = 0.05 / sqrt [ 0.00221 + 0.00194 ]
  = 0.05 / sqrt (0.00415)
  = 0.05 / 0.0644
  = -0.977

Therefore, the test statistic for testing the equality of the population proportions is -0.977, which corresponds to choice B.

Question 5

A random sample of 30 observations is selected from a normally distributed population. The sample variance is 12. In the 90% confidence interval for the population variance, the upper limit will be:

Accepted Answer

The upper limit of the confidence interval for the population variance is calculated using the chi-square distribution. For a 90% confidence interval with 29 degrees of freedom (n-1), the chi-square value for the upper limit is used. The formula is: (n-1)s^2/χ²(α/2), where s^2 is the sample variance. Plugging in the values gives the upper limit as approximately 19.652.

Question 6

Which of the following is a required condition for using the normal approximation to the binomial in constructing interval estimate for the difference between two population proportions?

Accepted Answer

The answer of Which of the following is a required...

Question 7

When the necessary conditions are met, a one-tail test is being conducted to test the difference between two population proportions, but your statistical software provides only a two-tail area of 0.058 as part of its output. The p-value for this test will be:

Accepted Answer

The answer of When the necessary conditions are met, a...

Question 8

In a hypothesis test for the population variance, the hypotheses are: $H _ { 0 } : \sigma ^ { 2 } = 175$ $H _ { 1 } : \sigma ^ { 2 } \neq 175$ If the sample size is 25 and the test is being carried out at the 5% level of significance, the rejection region will be:

Accepted Answer

The rejection region for a two-tailed chi-square test with 24 degrees of freedom (n-1) at the 5% significance level is determined by the critical values $\chi^2_{0.025, 24}$ and $\chi^2_{0.975, 24}$, which are approximately 12.4011 and 39.3641, respectively.

Question 9

Two samples of size 30 each 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 59.

Accepted Answer

The degrees of freedom for the equal-variances t-test statistic is calculated as the sum of the sample sizes minus 2, which is 30 + 30 - 2 = 58.

Question 10

Which of the following best describes a p-value?

Accepted Answer

A p-value is the probability of getting our sample results or more extreme if the null hypothesis about the population parameter were really true. It measures the strength of evidence against the null hypothesis. If the p-value is small (typically less than 0.05), we reject the null hypothesis and conclude that there is enough evidence to support the alternative hypothesis. If the p-value is large (typically greater than 0.05), we fail to reject the null hypothesis and do not have enough evidence to support the alternative hypothesis.

Question 11

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions. The two sample proportions are $p _ { 1 }$ = 0.32 and $p _ { 2 }$ = 0.38, and the standard error of the sampling distribution of $p _ { 1 } - p _ { 2 }$ is 0.046. The calculated value of the test statistic will be 1.3043.

Accepted Answer

The answer of When the necessary conditions are met, a...

Question 12

Which of the following is the number of degrees of freedom associated with the t-test, when the data are gathered from a matched pairs experiment with 30 pairs?

Accepted Answer

The number of degrees of freedom for a matched pairs t-test is equal to the number of pairs minus one, which in this case is 30-1=29. Therefore, the correct choice is B).

Question 13

When comparing two population variances, we test H₀: $\sigma _ { 1 } ^ { 2 } - \sigma _ { 2 } ^ { 2 }$ = 0.

Accepted Answer

The null hypothesis for comparing two population variances is typically stated as H0: $\sigma_1^2 = \sigma_2^2$, not as a difference of zero.

Question 14

In testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows: $n _ { 1 } = 20$ , $\bar { x } _ { 1 } = 10.8$ , $s _ { 1 } = 0.90$ . $n _ { 2 } = 18$ , $\bar { x } _ { 2 } = 9.6$ , $s _ { 2 } = 1.10$ . Assume that the population variances are unequal. The standard error of the sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ is equal to:

Accepted Answer

The standard error of the sampling distribution of the sample mean difference is:

$$
SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{(0.9)^2}{20} + \frac{(1.1)^2}{18}} \approx 0.3282
$$

Therefore, the correct answer is (B).

Question 15

From a sample of 500 items, 30 were found to be defective. The point estimate of the population proportion defective will be:

Accepted Answer

The point estimate of the population proportion defective is equal to the proportion of the sample that was found to be defective. In this case, 30 out of 500 items were defective, so the proportion defective is 30/500 = 0.06. Therefore, the point estimate of the population proportion defective is 0.06.

Question 16

If we reject a null hypothesis at the 0.05 level of significance, then we must also reject it at the 0.10 level.

Accepted Answer

If we reject the null hypothesis at the 0.05 level, it means that the probability of observing such extreme results or more extreme under the null hypothesis is less than 0.05. Therefore, if we increase the level of significance to 0.10, the probability of observing such extreme results or more extreme under the null hypothesis will be even less than 0.05. Thus, we will still reject the null hypothesis.

Question 17

In testing the hypotheses: $H _ { 0 } : p = 0.50$ $H _ { 1 } : p \neq 0.50$ , at the 10% significance level, if the sample proportion is 0.56, and the standard error of the sample proportion is 0.025, the appropriate conclusion is:

Accepted Answer

Using the formula for the z-test, we can calculate the test statistic:
$$z=\frac{\hat{p}-p}{\sqrt{p(1-p)/n}}=\frac{0.56-0.50}{0.025}=2.4$$
Since this is a two-tailed test, the critical value for a 10% significance level is approximately 1.645. Since the test statistic is greater than the critical value, we reject the null hypothesis. Therefore, the appropriate conclusion is to reject $H _ { 0 }$.

Question 18

For a sample of 25 observations taken from a normally distributed population with standard deviation of 6, a 95% confidence interval estimate for the population mean would require the use of:

Accepted Answer

Since the sample size is greater than 30, we can use a z-score to construct the confidence interval. A 95% confidence interval corresponds to a z-score of 1.96.

Question 19

Suppose that a one-tail t-test is being applied to find out if the population mean is at least 80. The level of significance is 0.10 and 25 observations were sampled. The rejection region is:

Accepted Answer

The answer of Suppose that a one-tail t-test is being...

Question 20

Two independent samples of sizes 35 and 40 are randomly selected from two normally distributed populations. Assume that the population variances are unknown but equal. In order to test the difference between the population means, $\mu _ { 1 } - \mu _ { 2 }$ , the sampling distribution of the sample mean difference, $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ , is:

Accepted Answer

The answer of Two independent samples of sizes 35 and...

Quiz 22: Statistical Inference: Conclusion

The pooled-variance estimator, $s _ { p } ^ { 2 }$ , requires that the two population variances be equal.

A one-tail test of the population proportion produces a test statistic z = -2.12. The p-value of the test is 0.034.

In testing the hypotheses: H₀ : $\mu$ = 140 H₁ : $\mu$ $\neq$ 140, Suppose that we rejected the null hypothesis at $\alpha$ = 0.10. Then for which of the following $\alpha$ values do we also reject the null hypothesis?

A sample of size 125 selected from one population has 55 successes, and a sample of size 140 selected from a second population has 70 successes. The test statistic for testing the equality of the population proportions is equal to:

A random sample of 30 observations is selected from a normally distributed population. The sample variance is 12. In the 90% confidence interval for the population variance, the upper limit will be:

Which of the following is a required condition for using the normal approximation to the binomial in constructing interval estimate for the difference between two population proportions?

When the necessary conditions are met, a one-tail test is being conducted to test the difference between two population proportions, but your statistical software provides only a two-tail area of 0.058 as part of its output. The p-value for this test will be:

In a hypothesis test for the population variance, the hypotheses are: $H _ { 0 } : \sigma ^ { 2 } = 175$ $H _ { 1 } : \sigma ^ { 2 } \neq 175$ If the sample size is 25 and the test is being carried out at the 5% level of significance, the rejection region will be:

Two samples of size 30 each 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 59.

Which of the following best describes a p-value?

Which of the following is the number of degrees of freedom associated with the t-test, when the data are gathered from a matched pairs experiment with 30 pairs?

When comparing two population variances, we test H₀: $\sigma _ { 1 } ^ { 2 } - \sigma _ { 2 } ^ { 2 }$ = 0.

From a sample of 500 items, 30 were found to be defective. The point estimate of the population proportion defective will be:

If we reject a null hypothesis at the 0.05 level of significance, then we must also reject it at the 0.10 level.

In testing the hypotheses: $H _ { 0 } : p = 0.50$ $H _ { 1 } : p \neq 0.50$ , at the 10% significance level, if the sample proportion is 0.56, and the standard error of the sample proportion is 0.025, the appropriate conclusion is:

For a sample of 25 observations taken from a normally distributed population with standard deviation of 6, a 95% confidence interval estimate for the population mean would require the use of:

Suppose that a one-tail t-test is being applied to find out if the population mean is at least 80. The level of significance is 0.10 and 25 observations were sampled. The rejection region is:

Quiz 22: Statistical Inference: Conclusion

The pooled-variance estimator, sp2s _ { p } ^ { 2 }sp2​ , requires that the two population variances be equal.

A one-tail test of the population proportion produces a test statistic z = -2.12. The p-value of the test is 0.034.

In testing the hypotheses: H0 : μ\muμ = 140 H1 : μ\muμ ≠\neq= 140, Suppose that we rejected the null hypothesis at α\alphaα = 0.10. Then for which of the following α\alphaα values do we also reject the null hypothesis?

A sample of size 125 selected from one population has 55 successes, and a sample of size 140 selected from a second population has 70 successes. The test statistic for testing the equality of the population proportions is equal to:

A random sample of 30 observations is selected from a normally distributed population. The sample variance is 12. In the 90% confidence interval for the population variance, the upper limit will be:

Which of the following is a required condition for using the normal approximation to the binomial in constructing interval estimate for the difference between two population proportions?

When the necessary conditions are met, a one-tail test is being conducted to test the difference between two population proportions, but your statistical software provides only a two-tail area of 0.058 as part of its output. The p-value for this test will be:

Two samples of size 30 each 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 59.

Which of the following best describes a p-value?

Which of the following is the number of degrees of freedom associated with the t-test, when the data are gathered from a matched pairs experiment with 30 pairs?

When comparing two population variances, we test H0: σ12−σ22\sigma _ { 1 } ^ { 2 } - \sigma _ { 2 } ^ { 2 }σ12​−σ22​ = 0.

From a sample of 500 items, 30 were found to be defective. The point estimate of the population proportion defective will be:

If we reject a null hypothesis at the 0.05 level of significance, then we must also reject it at the 0.10 level.

In testing the hypotheses: H0:p=0.50H _ { 0 } : p = 0.50H0​:p=0.50 H1:p≠0.50H _ { 1 } : p \neq 0.50H1​:p=0.50 , at the 10% significance level, if the sample proportion is 0.56, and the standard error of the sample proportion is 0.025, the appropriate conclusion is:

For a sample of 25 observations taken from a normally distributed population with standard deviation of 6, a 95% confidence interval estimate for the population mean would require the use of:

Suppose that a one-tail t-test is being applied to find out if the population mean is at least 80. The level of significance is 0.10 and 25 observations were sampled. The rejection region is:

The pooled-variance estimator, $s _ { p } ^ { 2 }$ , requires that the two population variances be equal.

In testing the hypotheses: H₀ : $\mu$ = 140 H₁ : $\mu$ $\neq$ 140, Suppose that we rejected the null hypothesis at $\alpha$ = 0.10. Then for which of the following $\alpha$ values do we also reject the null hypothesis?

When comparing two population variances, we test H₀: $\sigma _ { 1 } ^ { 2 } - \sigma _ { 2 } ^ { 2 }$ = 0.

In testing the hypotheses: $H _ { 0 } : p = 0.50$ $H _ { 1 } : p \neq 0.50$ , at the 10% significance level, if the sample proportion is 0.56, and the standard error of the sample proportion is 0.025, the appropriate conclusion is: