Question 1

Which of the following statements best describes a confidence interval for a population mean when the population standard deviation $$\sigma is $$known and the sample size $$n \geq 30$$?

Accepted Answer

It is an interval estimate that provides a range where the true population mean is likely to fall, calculated using the normal distribution.

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$$ with a known population standard deviation of $$\$$8,000. $$What is the 95% confidence interval for the population mean salary? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$\50$$,520$$ to $$\$$53,480$$

Question 3

Which formula should be used to calculate the margin of error for a confidence interval for the mean when $$\sigma is $$known?

Accepted Answer

$$ME = z_{\alpha/2} \dfrac{\sigma}{\sqrt{n}}$$

Question 4

If a 99% confidence interval for the mean is wider than a 90% confidence interval for the same data, what is the main reason for this?

Accepted Answer

A higher confidence level requires a larger critical value, increasing the margin of error.

Question 5

A hospital manager wants to estimate the average length of stay for patients. A sample of 15 patients has a mean of 6.2 days and a sample standard deviation of 1.1 days. What distribution and formula should be used to construct a 95% confidence interval?

Accepted Answer

Use the t-distribution: $$\bar{x} \pm t_{\alpha/2} \dfrac{s}{\sqrt{n}}$$

Question 6

Which of the following is a correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

We are 95% confident that the interval contains the true population mean.

Question 7

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

$$0.124 to$$ $$0.316$$

Question 8

Which condition must be met to use the normal approximation for a confidence interval for a population proportion?

Accepted Answer

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

Question 9

If the sample size increases while keeping the confidence level and population standard deviation constant, what happens to the margin of error?

Accepted Answer

The margin of error decreases.

Question 10

A sample of 35 flights has a mean of 145.5 tickets sold and a population standard deviation of 30.2. What is the standard error of the mean?

Accepted Answer

$$\dfrac{30.2}{\sqrt{35}} = 5.105$$

Question 11

Which formula is used to determine the required sample size $$n to $$estimate a population mean with a specified margin of error $$ME$$, confidence level, and known population standard deviation $$\sigma$$?

Accepted Answer

$$n = \left( \dfrac{z_{\alpha/2} \sigma}{ME} \$$right)^2$$

Question 12

A sample of 10 golf scores has a mean of 92.7 and a sample standard deviation of 4.52. What is the 90% confidence interval for the population mean? (Use $$t_{0.05,9} = 1.833)$$

Accepted Answer

$$90.1 to$$ $$95.3$$

Question 13

Which of the following best explains why the t-distribution is used instead of the normal distribution when constructing a confidence interval for the mean with unknown $$\sigma$$?

Accepted Answer

The t-distribution accounts for the extra uncertainty from estimating $$\sigma$$ with $$s$$ when the sample size is small.

Question 14

A pilot sample of 100 people found 22 with blue eyes. What sample size is needed to estimate the population proportion with 95% confidence and a margin of error of 5%? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$n = \dfrac{(1.96)^2$ (0.22)(0.78)}{(0.05)^2$} = 264$$

Question 15

Which of the following is the correct formula for the lower confidence limit (LCL) for the mean when using the finite population correction factor and $$\sigma is $$known?

Accepted Answer

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

Confidence Intervals in Business Statistics: Concepts, Calculations, and Applications

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