Q: A scientist wants to test if the average weight of a species of fish has changed from 5 kg. Which of the following is the correct set of hypotheses?

$$H_0$$: \mu = 5$$, H_1$: \mu \neq 5$$

Question 1

Which of the following formulas is used to calculate the margin of error for a confidence interval for a population mean when the population standard deviation is known?

Accepted Answer

$$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$

Question 2

A pharmaceutical company wants to estimate the proportion of patients who experience side effects from a new drug. Which formula should they use to determine the minimum sample size needed to estimate the population proportion with a specified margin of error $$E$$ and confidence level $$1-\alpha$$?

Accepted Answer

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

Question 3

Which of the following is the correct point estimate for a population mean $$\mu$$ based on a random sample?

Accepted Answer

The sample mean $$\bar{x}$$

Question 4

A researcher constructs a 95\% confidence interval for the mean systolic blood pressure in a population and obtains $$(120, 130). $$Which of the following statements is correct?

Accepted Answer

We are 95\% confident that the population mean lies between 120 and 130.

Question 5

Which of the following is the correct null hypothesis for testing whether the average commute time in a city is different from 30 minutes?

Accepted Answer

$$H_0$$: \mu = 30$$

Question 6

A hospital administrator wants to test if the proportion of patients satisfied with their care is less than 80\%. What is the correct alternative hypothesis?

Accepted Answer

$$H_1$$: p \u003c 0.80$$

Question 7

Which of the following best describes a Type I error in hypothesis testing?

Accepted Answer

Rejecting the null hypothesis when it is actually true.

Question 8

Given a sample mean $$\bar{x} = 52$$, population standard deviation $$\sigma = 10$$, sample size $$n = 25$$, and population mean $$\mu_0$ = 50$$, what is the value of the $$z-$$test statistic for testing $$H_0$$: \mu = 50$$?

Accepted Answer

$$z = \frac{52 - 50}{10/\sqrt{25}} = 1$$

Question 9

If the critical value for a right-tailed $$z-$$test at $$\alpha = 0.05 is$$ $$z_{0.05} = 1.645$$, which of the following $$z-$$test statistics would lead you to reject the null hypothesis?

Accepted Answer

$$z = 2.10$$

Question 10

A 99\% confidence interval for a population proportion is calculated as $$(0.45, 0.65). $$What is the correct interpretation?

Accepted Answer

We are 99\% confident that the true population proportion is between 0.45 and 0.65.

Question 11

Which of the following is the correct formula for the standard error of the sample proportion $$\hat{p}$$?

Accepted Answer

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

Question 12

A researcher wants to construct a 90\% confidence interval for a population mean using a sample of size $$n = 36$$ with sample mean $$\bar{x} = 100$$ and sample standard deviation $$s = 12. $$Which $$t-$$value should be used?

Accepted Answer

$$t_{0.05, 35}$$

Question 13

Which of the following statements is true about the relationship between confidence level and margin of error?

Accepted Answer

Increasing the confidence level increases the margin of error.

Question 14

A city council claims that at least 60\% of residents support a new recycling program. Which of the following is the correct null hypothesis for testing this claim?

Accepted Answer

$$H_0$$: p \geq 0.60$$

Question 15

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

Accepted Answer

The population mean must be known.

Question 16

A scientist wants to test if the average weight of a species of fish has changed from 5 kg. Which of the following is the correct set of hypotheses?

Accepted Answer

$$H_0$$: \mu = 5$$, H_1$: \mu 
eq 5$$

Question 17

Which of the following best describes the critical value in hypothesis testing?

Accepted Answer

The value that separates the rejection region from the non-rejection region.

Question 18

A sample of 100 voters shows that 58 support a candidate. Construct a 95\% confidence interval for the population proportion. Which formula should be used?

Accepted Answer

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

Question 19

Which of the following is the correct formula for the test statistic for a hypothesis test about a population proportion?

Accepted Answer

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

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