Q: A company wants to estimate the proportion of defective products in a large shipment. If a random sample of 200 products yields 16 defectives, what is the standard error of the sample proportion?

$$\sqrt{\frac{0.08 \times (1 - 0.08)}{200}}$$

Q: A medical researcher wants to estimate the mean blood pressure of adults in a city. The population standard deviation is known to be 12 mmHg. If a sample of 50 adults has a mean of 130 mmHg, what is the 95% confidence interval for the population mean?

$$130 \pm 1.96 \times \frac{12}{\sqrt{50}}$$

Q: A university wants to determine the sample size needed to estimate the average GPA of its students within a margin of error of 0.1, with 95% confidence. The population standard deviation is estimated to be 0.5. Which formula should be used?

$$n = \left( \frac{z_{\alpha/2} \times \sigma}{E} \$$right)^2$$

Q: A random sample of 40 employees has a mean salary of $$55,000. The population standard deviation is unknown, but the sample standard deviation is 8$$,000. What is the correct formula for the 99% confidence interval for the population mean?

$$55,000 \pm t_{0.005, 39} \times \frac{8,000}{\sqrt{40}}$$

Q: A manufacturer claims that less than 5% of its products are defective. A random sample of 100 products finds 7 defectives. What is the test statistic for testing the proportion at $$\alpha = 0.05$$?

$$z = \frac{0.07 - 0.05}{\sqrt{0.05 \times 0.95 / 100}}$$

Question 1

A company wants to estimate the proportion of defective products in a large shipment. If a random sample of 200 products yields 16 defectives, what is the standard error of the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.08 	imes (1 - 0.08)}{200}}$$

Question 2

Which of the following best describes the difference between a Type I error and a Type II error in hypothesis testing?

Accepted Answer

Type I error is rejecting a true null hypothesis, while Type II error is failing to reject a false null hypothesis.

Question 3

A medical researcher wants to estimate the mean blood pressure of adults in a city. The population standard deviation is known to be 12 mmHg. If a sample of 50 adults has a mean of 130 mmHg, what is the 95% confidence interval for the population mean?

Accepted Answer

$$130 \pm 1.96 	imes \frac{12}{\sqrt{50}}$$

Question 4

When estimating the population mean and the population standard deviation is unknown, which distribution should be used?

Accepted Answer

The t distribution

Question 5

A university wants to determine the sample size needed to estimate the average GPA of its students within a margin of error of 0.1, with 95% confidence. The population standard deviation is estimated to be 0.5. Which formula should be used?

Accepted Answer

$$n = \left( \frac{z_{\alpha/2} 	imes \sigma}{E} \$$right)^2$$

Question 6

In hypothesis testing, what does the p-value represent?

Accepted Answer

The probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.

Question 7

A business analyst tests whether the average time to process an order is less than 2 hours. Which of the following correctly states the null and alternative hypotheses?

Accepted Answer

$$H_0$$: \mu \geq 2$$; H_a$$: \mu \u003c 2$$

Question 8

A random sample of 40 employees has a mean salary of $$55,000. The population standard deviation is unknown, but the sample standard deviation is 8$$,000. What is the correct formula for the 99% confidence interval for the population mean?

Accepted Answer

$$55,000 \pm t_{0.005, 39} 	imes \frac{8,000}{\sqrt{40}}$$

Question 9

Which of the following statements about degrees of freedom in the t distribution is correct?

Accepted Answer

Degrees of freedom are equal to $$n - 1$$, where $$n is $$the sample size.

Question 10

A manufacturer claims that less than 5% of its products are defective. A random sample of 100 products finds 7 defectives. What is the test statistic for testing the proportion at $$\alpha = 0.05$$?

Accepted Answer

$$z = \frac{0.07 - 0.05}{\sqrt{0.05 	imes 0.95 / 100}}$$

Question 11

If the level of significance $$\alpha is $$set at 0.01, what does this imply about the probability of a Type I error?

Accepted Answer

There is a 1% chance of rejecting a true null hypothesis.

Question 12

A retail store wants to estimate the proportion of customers who use credit cards. What is the correct formula for the 95% confidence interval for the population proportion?

Accepted Answer

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

Question 13

A pharmaceutical company wants to determine the sample size needed to estimate the proportion of patients who respond to a new drug within a margin of error of 0.03, with 90% confidence. The estimated proportion is 0.6. Which formula should be used?

Accepted Answer

$$n = \frac{z_{\alpha/2}^2 	imes \hat{p} (1 - \hat{p})}{E^2$}$$

Question 14

A one-tailed hypothesis test is being conducted to determine if the mean weight of cereal boxes is less than 500g. Which rejection region is appropriate?

Accepted Answer

$$z \u003c -z_{\alpha}$$

Question 15

A sample of 25 students has a mean test score of 78 and a sample standard deviation of 10. If you want to test whether the population mean is different from 75 at $$\alpha = 0.05$$, which test should you use?

Accepted Answer

A t-test for the mean

Question 16

Which of the following is NOT a requirement for using the z-test for the population mean?

Accepted Answer

The population standard deviation is unknown

Question 17

A hospital administrator wants to test if the proportion of patients who recover within 3 days is greater than 0.7. Which is the correct alternative hypothesis?

Accepted Answer

$$H_a: p \u003e 0.7$$

Question 18

A sample of 60 cars has an average fuel efficiency of 25 mpg. The population standard deviation is known to be 4 mpg. What is the z-statistic for testing if the population mean is 24 mpg?

Accepted Answer

$$z = \frac{25 - 24}{4 / \sqrt{60}}$$

Sampling Distributions, Confidence Intervals, and One-Sample Hypothesis Testing: Exam 3 Study Guide

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