Question 1

A pharmaceutical company wants to estimate the average time it takes for a new drug to take effect. The population variance is known. Which formula should they use to construct a 95% confidence interval for the mean effect time?

Accepted Answer

$$ X \pm Z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} $$

Question 2

Which of the following factors will result in a narrower confidence interval for the mean?

Accepted Answer

Increasing the sample size $$n$$

Question 3

A market researcher wants to estimate the proportion of customers who prefer a new product. Which formula should they use for the confidence interval estimate?

Accepted Answer

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

Question 4

If the population variance is unknown, which distribution should be used to construct a confidence interval for the mean?

Accepted Answer

Student's t distribution

Question 5

A company wants to estimate the mean salary of its employees with a margin of error $$e. $$The population variance is known. Which formula should they use to determine the required sample size?

Accepted Answer

$$ n = \frac{Z_{\frac{\alpha}{2}}^2 \$$sigma^2$$}{e^2$} $$

Question 6

Which of the following statements best describes the interpretation of a 95% confidence interval for the mean?

Accepted Answer

If all possible samples of size $$n$$ are taken, 95% of the intervals will contain the true population mean.

Question 7

A survey of 200 students found that 60 prefer online classes. What is the point estimate for the proportion of students who prefer online classes?

Accepted Answer

$$ \frac{60}{200} = 0.3 $$

Question 8

Which assumption is necessary for using the normal approximation in confidence interval estimation for a proportion?

Accepted Answer

Both $$np \geq 5$$ and $$n(1-p) \geq 5$$

Question 9

A business analyst wants to estimate the proportion of defective products in a batch. If the acceptable sampling error is $$e$$ and the estimated proportion is $$p$$, which formula should they use to determine the required sample size?

Accepted Answer

$$ n = \frac{Z_{\frac{\alpha}{2}}^2 p(1-p)}{e^2$} $$

Question 10

Which of the following will increase the width of a confidence interval for the mean?

Accepted Answer

Increasing the level of confidence

Question 11

A hospital administrator wants to estimate the average waiting time for patients. The sample mean is 30 minutes, the sample standard deviation is 8 minutes, and the sample size is 25. The population variance is unknown. What is the correct formula for the 95% confidence interval?

Accepted Answer

$$ X \pm t_{\frac{\alpha}{2}, n-1} \frac{S}{\sqrt{n}} $$

Question 12

Which of the following is NOT an element of a confidence interval estimate?

Accepted Answer

Population size

Question 13

A company wants to estimate the mean number of sales calls made per week. They want a 99% confidence interval. How will the interval width compare to a 95% confidence interval, all else equal?

Accepted Answer

The 99% confidence interval will be wider than the 95% confidence interval.

Question 14

Which of the following is a correct statement about confidence intervals?

Accepted Answer

A confidence interval provides a range of values likely to contain the population parameter.

Question 15

A quality control manager wants to estimate the proportion of defective items. If the sample proportion is 0.05 and the sample size is 400, what is the standard error for the proportion?

Accepted Answer

$$ \sqrt{\frac{0.05 	imes 0.95}{400}} $$

Question 16

Which of the following is NOT an assumption for constructing a confidence interval estimate for the mean when the population variance is unknown?

Accepted Answer

Population variance is known

Question 17

A researcher wants to estimate the mean weight of apples in an orchard. The sample mean is 150g, sample standard deviation is 10g, and sample size is 36. The population variance is unknown. What is the margin of error for a 90% confidence interval?

Accepted Answer

$$ t_{0.05, 35} 	imes \frac{10}{\sqrt{36}} $$

Question 18

A business wants to estimate the proportion of customers who return products. If the estimated proportion is 0.2, the desired margin of error is 0.05, and the confidence level is 95%, what is the required sample size?

Accepted Answer

$$ n = \frac{Z_{0.025}^2 	imes 0.2 	imes 0.8}{0.05^2$} $$

Question 19

A confidence interval for the mean is calculated as $$ (45, 55) . $$What is the point estimate for the mean?

Accepted Answer

$$ \frac{45 + 55}{2} = 50 $$

Chapter 8: Confidence Interval Estimation – Study Notes

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