Question 1

A pharmaceutical company wants to estimate the mean recovery time for a new drug. A sample of 36 patients has a mean recovery time of 8 days with a population standard deviation of 3 days. Construct a 95% confidence interval for the mean recovery time. Which formula should be used?

Accepted Answer

Use $$\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$$

Question 2

Which of the following is the correct test statistic for testing the difference between two population proportions?

Accepted Answer

$$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1$} + \frac{1}{n_2$}\right)}}$$

Question 3

A researcher wants to test if the average commute time in a city is different from 30 minutes. A random sample of 25 commuters has a mean of 32 minutes and a sample standard deviation of 5 minutes. Which test statistic should be used?

Accepted Answer

$$t = \frac{\bar{x} - \mu_0$}{\frac{s}{\sqrt{n}}}$$

Question 4

Which formula is used to construct a confidence interval for a population proportion?

Accepted Answer

$$p \pm z_{\alpha/2} \sqrt{\frac{p(1-p)}{n}}$$

Question 5

A hospital administrator wants to compare the mean waiting times in two departments. The population variances are known. Which formula should be used for the confidence interval?

Accepted Answer

$$\mu_1$ - \mu_2$ \pm z_{\alpha/2} \sqrt{\frac{\$$sigma_1$$^2}{n_1$} + \frac{\$$sigma_2$$^2}{n_2$}}$$

Question 6

Which formula is appropriate for a hypothesis test for a single population mean when the population standard deviation is unknown?

Accepted Answer

$$t = \frac{\bar{x} - \mu_0$}{\frac{s}{\sqrt{n}}}$$

Question 7

A company claims that at least 80% of its products pass quality control. A sample of 200 products shows 150 pass. What is the test statistic for testing the company's claim?

Accepted Answer

$$z = \frac{p - p_0$}{\sqrt{\frac{p_0$(1-p_0$)}{n}}}$$

Question 8

When constructing a confidence interval for the difference between two means with unknown but equal variances, which formula should be used?

Accepted Answer

$$\mu_1$ - \mu_2$ \pm t_{\alpha/2} s_p \sqrt{\frac{1}{n_1$} + \frac{1}{n_2$}}$$

Question 9

A researcher is testing $$H_0$$: \mu = 100$$ vs H_1$: \mu 
eq 100$$ with a sample mean of 104, sample standard deviation of 8, and sample size of 16. What is the value of the test statistic?

Accepted Answer

$$t = \frac{104 - 100}{\frac{8}{\sqrt{16}}} = 2$$

Question 10

Which of the following best describes the critical value $$z_{\alpha/2}$$ for a 98% confidence interval?

Accepted Answer

$$z_{\alpha/2} is $$the value such that the area to the right is $$0.01$$

Question 11

A study compares the proportion of smokers in two cities. Which formula should be used for the confidence interval for the difference in proportions?

Accepted Answer

$$p_1$$ - $$p_2$$ \pm z_{\alpha/2} \sqrt{\frac{$$p_1(1$$-$$p_1$$)}{$$n_1$$} + \frac{$$p_2(1$$-$$p_2$$)}{$$n_2$$}}$$

Question 12

Which formula is used to find the critical value for a left-tailed hypothesis test at significance level $$\alpha$$?

Accepted Answer

$$z_{\alpha} is $$found such that the area to the left is $$\alpha$$

Question 13

A sample of 50 batteries has a mean life of 100 hours and a known population standard deviation of 10 hours. What is the 90% confidence interval for the mean battery life?

Accepted Answer

$$100 \pm 1.645 	imes \frac{10}{\sqrt{50}}$$

Question 14

Which formula is used for the test statistic when comparing two means from independent samples with unknown and unequal variances?

Accepted Answer

$$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}}$$

Question 15

A left-tailed test is being conducted at $$\alpha = 0.05. $$What is the critical value $$z_{\alpha}$$?

Accepted Answer

$$z_{\alpha} = -1.645$$

Question 16

Which formula is used for the test statistic for a hypothesis test for a single population proportion?

Accepted Answer

$$z = \frac{p - p_0$}{\sqrt{\frac{p_0$(1-p_0$)}{n}}}$$

Confidence Intervals and Hypothesis Testing: Key Formulas and Procedures

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