Question 1

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 2

A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim against the alternative that the actual proportion of doctors who recommend aspirin is less than 0.90, a random sample of 100 doctors' results in 83 who indicate that they recommend aspirin. The value of the test statistic in this problem is approximately equal to:

Accepted Answer

The null hypothesis is that 90% of doctors recommend aspirin, so the probability of a doctor recommending aspirin is 0.9. The alternative hypothesis is that the proportion of doctors who recommend aspirin is less than 0.90.

The sample size is n=100, the number of doctors who recommend aspirin is x=83. So, the sample proportion is $\hat p = \frac{x}{n} = \frac{83}{100} = 0.83$.

Under the null hypothesis, the test statistic is approximately standard normal, so we can use the standard normal distribution to find the p-value. The test statistic is:

$z = \frac{\hat p - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} = \frac{0.83-0.90}{\sqrt{\frac{0.9	imes0.1}{100}}} \approx -2.33$

Using a standard normal table or calculator, the p-value is approximately 0.0107. Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the actual proportion of doctors who recommend aspirin is less than 0.90.

The best choice is B because it is the closest approximation of the test statistic.

Question 3

After calculating the sample size needed to estimate a population proportion to within 0.04, your statistics professor told you the maximum allowable error must be reduced to just .01. If the original calculation led to a sample size of 800, the sample size will now have to be:

Accepted Answer

The sample size needed to estimate a population proportion with a certain level of confidence is inversely proportional to the square of the allowable error. If the allowable error decreases from 0.04 to 0.01, the square of the ratio of the original error to the new error is (0.04/0.01)^2 = 16. Therefore, the new sample size is 16 times the original sample size: 800 * 16 = 12,800.

Question 4

If you wish to test whether two populations means are the same, the appropriate null and alternative hypotheses would be:  vs.  .

Accepted Answer

The appropriate null hypothesis would be that the two population means are equal (11eb6b9e_0639_d436_8d9d_4d2034de2c3c_TB8220_11) and the alternative hypothesis would be that they are not equal (11eb6b9e_0639_d437_8d9d_17d375d7bc5f_TB8220_11).

Question 5

The z-test can be used to determine whether two population means are equal.

Accepted Answer

The answer of The z-test can be used to determine...

Question 6

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 with normal populations and equal variances, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference.

Question 7

In estimating the difference between two population means, the following summary statistics were found:      and  Based on these results, the point estimate of  is .70.

Accepted Answer

The summary statistics provided do not include enough information to calculate the point estimate of the difference between two population means.

Question 8

The width of a confidence interval estimate for a proportion will be:

Accepted Answer

The width of a confidence interval is narrower for a lower confidence level (90%) compared to a higher confidence level (95%) because a lower confidence level means you are accepting a higher chance of error, thus requiring less range to capture the true parameter.

Question 9

Which of the following would be an appropriate null hypothesis?

Accepted Answer

The null hypothesis should always be stated in terms of the population parameter, which in this case is the population proportion. Option A correctly states that the population proportion is equal to a specific value (0.60), making it an appropriate null hypothesis. Options B and C are incorrect because they refer to the sample proportion instead of the population proportion. Option D is incorrect because it specifies a one-tailed alternative hypothesis.

Question 10

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 political analyst is surveying two different populations (registered Democrats and registered Republicans) and comparing the results, which makes this an example of two independent samples.

Question 11

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 be not to reject the null hypothesis.

Accepted Answer

At a 5% significance level for a two-tailed test, the critical values for a z-test (assuming the population standard deviations are known, thus a z-test applies) are approximately ±1.96. A test statistic of 2.75 exceeds this critical value, indicating that the null hypothesis should be rejected.

Question 12

The significance level in a hypothesis test for the difference between two population means is the same as the probability of committing a Type I error.

Accepted Answer

The significance level (alpha) is the probability of rejecting the null hypothesis when it is actually true, which is equivalent to committing a Type I error. Therefore, the significance level is the same as the probability of committing a Type I error.

Question 13

With all other factors held constant, increasing the confidence level for a confidence interval estimate for the difference between two population means will result in a wider confidence interval estimate.

Accepted Answer

Increasing the confidence level requires a higher level of certainty, which means allowing for a larger margin of error. This results in a wider interval estimate.

Question 14

Which of the following would be an appropriate alternative hypothesis?

Accepted Answer

The alternative hypothesis should state a specific deviation from the null hypothesis. A is the only choice that does this by stating that the population proportion is less than 0.65, which is different from the null hypothesis of the population proportion being equal to 0.65. B and D simply reference the sample proportion which is not the parameter of interest in hypothesis testing. C is the same as the null hypothesis and does not state a deviation from it. E is not specific enough as it includes both a deviation from the null hypothesis and a reference to the sample proportion.

Question 15

If we reject the null hypothesis  at the 0.01 level of significance, then we must also reject it at the 0.05 level.

Accepted Answer

If we reject the null hypothesis at a lower level of significance (e.g. 0.01), it means that the evidence against the null hypothesis is stronger. Therefore, we would also reject the null hypothesis at a higher level of significance (e.g. 0.05).

Question 16

From a sample of 400 items, 14 are found to be defective. The point estimate of the population proportion defective will be:

Accepted Answer

The point estimate of the population proportion defective can be calculated as:
point estimate = number of defective items / sample size
point estimate = 14 / 400
point estimate = 0.035
Therefore, the best choice is B.

Question 17

Independent samples are those for which the selection process for one is not related to the selection process for the other.

Accepted Answer

Independent samples are defined as samples that are not related to each other in terms of the selection process. This means that the selection of one sample does not affect the selection of the other sample.

Question 18

Increasing the size of the samples in a study to estimate the difference between two population means will increase the probability of committing a Type I error that a decision maker can have regarding the interval estimate.

Accepted Answer

Increasing the sample size actually decreases the probability of committing a Type I error (rejecting a true null hypothesis). As the sample size grows larger, the estimate becomes more precise and the margin of error decreases, leading to a narrower interval estimate and a more accurate conclusion.

Question 19

In estimating the difference between two population means, if a 90% confidence interval includes zero, then we can be 90% certain that the difference between the two population means is zero.

Accepted Answer

If the confidence interval includes zero, it means that there is a possibility that the difference between the two means is zero. However, we cannot be certain that the difference is actually zero.

Question 20

In estimating the difference between two population means, the estimate for the standard deviation of the sampling distribution of  is found by taking the square root of the sum of the two sample variances.

Accepted Answer

The estimate for the standard deviation of the sampling distribution of the difference between two population means is found by taking the square root of the sum of the two population variances divided by the sample sizes.

Quiz 9: Large-Sample Tests of Hypotheses

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.