Q: A researcher wants to construct a 95% confidence interval for the mean cholesterol level in adults. The sample mean is 210 mg/dL, the sample standard deviation is 30 mg/dL, and the sample size is 36. What is the margin of error?

$$ME = z^* \times \frac{s}{\sqrt{n}} = 1.96 \times \frac{30}{6} = 1.96 \times 5 = 9.8$$

Question 1

A pharmaceutical company claims that 60% of patients experience relief after taking a new drug. In a clinical trial, 120 out of 200 patients reported relief. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{120}{200} = 0.60$$

Question 2

Which formula is used to calculate the standard error of the sample proportion $$\hat{p}$$ when the population proportion $$p is $$known?

Accepted Answer

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

Question 3

A researcher wants to construct a 95% confidence interval for the mean cholesterol level in adults. The sample mean is 210 mg/dL, the sample standard deviation is 30 mg/dL, and the sample size is 36. What is the margin of error?

Accepted Answer

$$ME = z^* 	imes \frac{s}{\sqrt{n}} = 1.96 	imes \frac{30}{6} = 1.96 	imes 5 = 9.8$$

Question 4

Which of the following statements about hypothesis testing is correct?

Accepted Answer

Failing to reject the null hypothesis does NOT mean the null hypothesis is true.

Question 5

A hospital wants to compare the proportion of patients who recover within a week between two treatments. Treatment A: 45 out of 60 recover; Treatment B: 30 out of 50 recover. What is the difference in sample proportions $$\hat{p}_1 - \hat{p}_2$$?

Accepted Answer

$$\hat{p}_1 - \hat{p}_2 = \frac{45}{60} - \frac{30}{50} = 0.75 - 0.60 = 0.15$$

Question 6

Which formula is used to calculate the test statistic for a single population mean when the population standard deviation is unknown?

Accepted Answer

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

Question 7

A scientist wants to test if the average weight of a species of fish is different from 5 kg. She collects a sample of 25 fish, finds a sample mean of 5.3 kg and a sample standard deviation of 0.8 kg. What is the value of the test statistic?

Accepted Answer

$$t = \frac{5.3 - 5}{0.8/\sqrt{25}} = \frac{0.3}{0.16} = 1.875$$

Question 8

When constructing a confidence interval for a population mean using the $$t-$$distribution, which condition must be met?

Accepted Answer

The sample must be drawn from a population that is approximately normal if the sample size is small.

Question 9

A company wants to estimate the proportion of defective products in a shipment. They sample 150 items and find 12 are defective. Construct a 95% confidence interval for the proportion of defective items. What is the correct formula for the interval?

Accepted Answer

$$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 10

In hypothesis testing for two population proportions, which is the correct null hypothesis?

Accepted Answer

$$H_0$$: $$p_1$$ - $$p_2 = 0$

Question 11

A medical researcher wants to test if the mean blood pressure in a population is less than 120 mmHg. Which of the following is the correct alternative hypothesis?

Accepted Answer

$$H_a: \mu \u003c 120$$

Question 12

Which of the following best describes the p-value in hypothesis testing?

Accepted Answer

The probability of observing a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.

Question 13

A researcher calculates a 99% confidence interval for the mean height of a plant species and finds it to be (15.2 cm, 18.8 cm). What does this interval mean?

Accepted Answer

We are 99% confident that the true mean height of the plant species is between 15.2 cm and 18.8 cm.

Question 14

Which formula is used to calculate the test statistic for the difference in two population proportions?

Accepted Answer

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

Question 15

A scientist wants to test if the mean reaction time for a new drug is different from the standard reaction time of 2.5 seconds. She collects a sample and performs a two-tailed $$t-$$test. What is the correct alternative hypothesis?

Accepted Answer

$$H_a: \mu 
eq 2.5$$

Question 16

Which of the following is NOT a requirement for constructing a confidence interval for a population proportion?

Accepted Answer

The sample size must be less than 10.

Question 17

A researcher performs a hypothesis test and obtains a p-value of 0.03. If the significance level is $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is evidence for the alternative hypothesis.

Question 18

Which formula is used to calculate the confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

$$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$$

Study Guide: Inference for Means and Proportions (Chapters 16–20)

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