Q: A company wants to estimate the proportion of defective products. What sample size is needed to estimate the population proportion with a margin of error of 0.05 at a 95% confidence level, assuming no pilot sample is available? (Use $$z_{0.025} = 1.96)$$

$$n = \frac{(1.96)^2$ \times 0.5 \times 0.5}{(0.05)^2$}$$

Question 1

Which of the following best describes the purpose of a confidence interval in business statistics?

Accepted Answer

To give a range of values that is likely to contain the population parameter with a certain level of confidence.

Question 2

A company wants to estimate the average salary of its employees. A random sample of 40 employees yields a sample mean of $$52,000 and a known population standard deviation of 8$$,000. What is the 95% confidence interval for the population mean salary? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$52,000 \pm 2,478.4$$

Question 3

Which formula should be used to calculate the confidence interval for the mean when the population standard deviation is unknown and the sample size is less than 30?

Accepted Answer

$$\bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}}$$

Question 4

A hospital manager wants to estimate the average number of days patients spend in the hospital. A sample of 15 patients has a mean of 5.11 days and a sample standard deviation of 0.85 days. What is the 95% confidence interval for the population mean? (Use $$t_{0.025,14} = 2.145)$$

Accepted Answer

$$5.11 \pm 0.4707$$

Question 5

Which of the following statements about the margin of error in a confidence interval is correct?

Accepted Answer

Increasing the sample size decreases the margin of error.

Question 6

A random sample of 100 people found that 22 have blue eyes. What is the 98% confidence interval for the population proportion of blue-eyed people? (Use $$z_{0.01} = 2.33)$$

Accepted Answer

$$0.22 \pm 0.096$$

Question 7

Which condition must be met to use the normal distribution to approximate the binomial distribution for confidence intervals of proportions?

Accepted Answer

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

Question 8

A sample of 35 luxury hotels has an average room rate of $$247.80 and a population standard deviation of 39. $$What is the standard error of the mean?

Accepted Answer

$$6.59$$

Question 9

If a 95% confidence interval for a mean is calculated as (20, 30), which of the following is a correct interpretation?

Accepted Answer

We are 95% confident that the interval from 20 to 30 contains the true population mean.

Question 10

Which formula is used to determine the required sample size to estimate a population mean with a specified margin of error $$ME$$ and confidence level?

Accepted Answer

$$n = \frac{z_{\alpha/2}^2 \$$sigma^2$$}{ME^2$}$$

Question 11

A business analyst is constructing a confidence interval for the mean using a finite population correction factor. Which formula should be used for the upper confidence limit when the population standard deviation is known?

Accepted Answer

$$UCL_{\bar{x}} = \bar{x} + z_{\alpha/2} \sigma_{\bar{x}} \left( \sqrt{\frac{N-n}{N-1}} \right)$$

Question 12

Which of the following is NOT an assumption required when constructing a confidence interval for the mean using the t-distribution?

Accepted Answer

The population standard deviation is known.

Question 13

A sample of 500 car deals found that 98 used 0% financing. Construct a 98% confidence interval for the proportion of 0% financing deals. (Use $$z_{0.01} = 2.33$$ and $$\hat{p} = 0.196)$$

Accepted Answer

$$0.196 \pm 0.041$$

Question 14

Which of the following best explains why the t-distribution is used instead of the normal distribution when the population standard deviation is unknown?

Accepted Answer

The t-distribution accounts for the additional uncertainty from estimating the population standard deviation with the sample standard deviation.

Question 15

A company wants to estimate the proportion of defective products. What sample size is needed to estimate the population proportion with a margin of error of 0.05 at a 95% confidence level, assuming no pilot sample is available? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$n = \frac{(1.96)^2$ 	imes 0.5 	imes 0.5}{(0.05)^2$}$$

Question 16

If the confidence level is increased from 90% to 99%, what happens to the width of the confidence interval, assuming all other factors remain constant?

Accepted Answer

The width of the confidence interval increases.

Chapter 8

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