Q: A company wants to estimate the proportion of its customers who are satisfied with their service. If a random sample of 400 customers yields a sample proportion $$\hat{p} = 0.65$$, what is the standard error of the sample proportion?

$$\sqrt{\dfrac{0.65 \times 0.35}{400}}$$

Q: A hospital administrator wants to estimate the average length of stay for patients. The population standard deviation $$\sigma is $$known to be 2.5 days. A random sample of 64 patients has a mean stay of 5.2 days. What is the 95% confidence interval for the population mean?

$$5.2 \pm 1.96 \times \dfrac{2.5}{8}$$

Q: A researcher wants to determine the sample size needed to estimate a population mean within $$\pm 1$$ unit with 95% confidence. The population standard deviation is estimated to be 4. What is the minimum sample size required?

$$\left( \dfrac{1.96 \times 4}{1} \$$right)^2$$

Q: A sample of 25 students has a mean test score of 78. The population standard deviation is unknown, and the sample standard deviation is 10. What is the correct formula for the 99% confidence interval for the population mean?

$$78 \pm t_{0.005,24} \times \dfrac{10}{\sqrt{25}}$$

Q: A business analyst is testing whether the proportion of defective products is different from 5%. Which of the following is the correct set of hypotheses?

$$H_0$$: p = 0.05$$; H_a$$: p \neq 0.05$$

Q: A sample of 100 voters shows that 60% support a new policy. What is the 90% confidence interval for the population proportion?

$$0.6 \pm 1.645 \times \sqrt{\dfrac{0.6 \times 0.4}{100}}$$

Question 1

A company wants to estimate the proportion of its customers who are satisfied with their service. If a random sample of 400 customers yields a sample proportion $$\hat{p} = 0.65$$, what is the standard error of the sample proportion?

Accepted Answer

$$\sqrt{\dfrac{0.65 	imes 0.35}{400}}$$

Question 2

A hospital administrator wants to estimate the average length of stay for patients. The population standard deviation $$\sigma is $$known to be 2.5 days. A random sample of 64 patients has a mean stay of 5.2 days. What is the 95% confidence interval for the population mean?

Accepted Answer

$$5.2 \pm 1.96 	imes \dfrac{2.5}{8}$$

Question 3

Which of the following statements about the t-distribution is TRUE?

Accepted Answer

As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.

Question 4

A researcher wants to determine the sample size needed to estimate a population mean within $$\pm 1$$ unit with 95% confidence. The population standard deviation is estimated to be 4. What is the minimum sample size required?

Accepted Answer

$$\left( \dfrac{1.96 	imes 4}{1} \$$right)^2$$

Question 5

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

Accepted Answer

Failing to reject the null hypothesis when the alternative hypothesis is true.

Question 6

A manufacturer claims that the mean lifetime of its light bulbs is at least 1,000 hours. To test this claim, what are the correct null and alternative hypotheses?

Accepted Answer

$$H_0$$: \mu = 1000$$; H_a$$: \mu \u003c 1000$$

Question 7

A sample of 25 students has a mean test score of 78. The population standard deviation is unknown, and the sample standard deviation is 10. What is the correct formula for the 99% confidence interval for the population mean?

Accepted Answer

$$78 \pm t_{0.005,24} 	imes \dfrac{10}{\sqrt{25}}$$

Question 8

A business analyst is testing whether the proportion of defective products is different from 5%. Which of the following is the correct set of hypotheses?

Accepted Answer

$$H_0$$: p = 0.05$$; H_a$$: p 
eq 0.05$$

Question 9

In a one-tailed hypothesis test with $$\alpha = 0.05$$, the calculated test statistic is $$z = 1.8. $$What is the correct decision?

Accepted Answer

Fail to reject the null hypothesis.

Question 10

A sample of 100 voters shows that 60% support a new policy. What is the 90% confidence interval for the population proportion?

Accepted Answer

$$0.6 \pm 1.645 	imes \sqrt{\dfrac{0.6 	imes 0.4}{100}}$$

Question 11

Which of the following is NOT a requirement for constructing a confidence interval for a population mean when $$\sigma is $$unknown?

Accepted Answer

The population standard deviation must be known.

Question 12

A pharmaceutical company wants to determine the sample size needed to estimate the proportion of patients experiencing side effects within $$\pm 0.03$$ with 95% confidence. If no prior estimate is available, what value of $$p$$ should be used in the sample size formula?

Accepted Answer

$$p = 0.5$$

Question 13

A hypothesis test is conducted at the 1% significance level ($$\alpha = 0.01). $$The p-value is calculated to be 0.008. What is the correct conclusion?

Accepted Answer

Reject the null hypothesis.

Question 14

Which of the following best describes the standard error of the sample mean when $$\sigma is $$known?

Accepted Answer

$$\dfrac{\sigma}{\sqrt{n}}$$

Question 15

A one-tailed t-test is performed with a sample size of 16. What are the degrees of freedom for the test statistic?

Accepted Answer

$$15$$

Question 16

A company wants to test if the mean delivery time is less than 3 days. A sample of 36 deliveries has a mean of 2.8 days and a known standard deviation of 0.6 days. What is the value of the test statistic?

Accepted Answer

$$\dfrac{2.8 - 3}{0.6/6}$$

Question 17

Which of the following is the correct formula for the sample size needed to estimate a population proportion with margin of error $$E$$ and confidence level $$z_{\alpha/2}$$, using an estimated proportion $$\hat{p}$$?

Accepted Answer

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

Question 18

A bank wants to test if the average waiting time is different from 10 minutes. A random sample of 50 customers has a mean waiting time of 9.5 minutes and a known standard deviation of 2 minutes. What is the null hypothesis?

Accepted Answer

$$H_0$$: \mu = 10$$

Question 19

Which of the following is TRUE about the level of confidence in a confidence interval?

Accepted Answer

A higher level of confidence results in a wider confidence interval.

Sampling Distributions, Confidence Intervals, and One-Sample Hypothesis Testing

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