Q: Suppose a sample of 15 patients has a mean hospital stay of 5.11 days and a sample standard deviation of 0.85 days. Calculate the 95% confidence interval for the population mean hospital stay. Use the critical t-score $$t_{0.025, 14} = 2.145.$$

$$5.11 \pm (2.145 \times \frac{0.85}{\sqrt{15}})$$, which is approximately (4.64, 5.58)

Q: Given a sample proportion $$p = 0.22$$ from a sample of $$n = 100$$, calculate the standard error of the proportion.

$$SE = \sqrt{\frac{0.22 \times 0.78}{100}} \approx 0.0414$$

Q: A survey of 1,163 people finds that 40% prefer football. If the standard error is 0.014, what is the 95% confidence interval for the proportion preferring football? (Use $$z_{0.025} = 1.96)$$

$$0.40 \pm (1.96 \times 0.014)$$, which is (0.3736, 0.4264)

Q: A business analyst wants to estimate the proportion of customers satisfied with a new product. In a sample of 200 customers, 160 report satisfaction. What is the 99% confidence interval for the proportion satisfied? (Use $$z_{0.005} = 2.575)$$

$$0.80 \pm (2.575 \times \sqrt{\frac{0.8 \times 0.2}{200}})$$, which is approximately (0.737, 0.863)

Question 1

Which of the following best describes a point estimate in statistics?

Accepted Answer

A single value calculated from sample data that best describes the population parameter of interest.

Question 2

Why is it important to report a confidence interval along with a point estimate when presenting survey results, such as unemployment rates?

Accepted Answer

Because point estimates alone do not provide information about their accuracy or the uncertainty due to sampling error.

Question 3

Suppose a sample of 15 patients has a mean hospital stay of 5.11 days and a sample standard deviation of 0.85 days. Calculate the 95% confidence interval for the population mean hospital stay. Use the critical t-score $$t_{0.025, 14} = 2.145.$$

Accepted Answer

$$5.11 \pm (2.145 	imes \frac{0.85}{\sqrt{15}})$$, which is approximately (4.64, 5.58)

Question 4

Which formula is used to calculate the standard error of the mean when the population standard deviation $$\sigma is $$unknown?

Accepted Answer

$$SE = \frac{s}{\sqrt{n}}$$

Question 5

A 95% confidence interval for the mean is interpreted as:

Accepted Answer

If we repeated the sampling process infinitely, 95% of the calculated intervals would contain the true population mean.

Question 6

Which of the following factors will NOT increase the width of a confidence interval for the mean?

Accepted Answer

Increasing the sample size $$n$$

Question 7

When constructing a confidence interval for a population proportion, which condition must be met to use the normal approximation?

Accepted Answer

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

Question 8

Given a sample proportion $$p = 0.22$$ from a sample of $$n = 100$$, calculate the standard error of the proportion.

Accepted Answer

$$SE = \sqrt{\frac{0.22 	imes 0.78}{100}} \approx 0.0414$$

Question 9

Which Excel function would you use to find the critical t-score for a 95% confidence interval with 14 degrees of freedom?

Accepted Answer

T.INV.2T(0.05, 14)

Question 10

A survey of 1,163 people finds that 40% prefer football. If the standard error is 0.014, what is the 95% confidence interval for the proportion preferring football? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$0.40 \pm (1.96 	imes 0.014)$$, which is (0.3736, 0.4264)

Question 11

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

Accepted Answer

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

Question 12

If you want to decrease the margin of error in a confidence interval for a mean, which action would be most effective?

Accepted Answer

Increase the sample size $$n$$

Question 13

A business analyst wants to estimate the proportion of customers satisfied with a new product. In a sample of 200 customers, 160 report satisfaction. What is the 99% confidence interval for the proportion satisfied? (Use $$z_{0.005} = 2.575)$$

Accepted Answer

$$0.80 \pm (2.575 	imes \sqrt{\frac{0.8 	imes 0.2}{200}})$$, which is approximately (0.737, 0.863)

Question 14

Which of the following is the correct formula for the margin of error in a confidence interval for a proportion?

Accepted Answer

$$ME = z_{\alpha/2} 	imes \sqrt{\frac{p(1-p)}{n}}$$

Question 15

A company wants to estimate the average time customers spend on its website. The population standard deviation is known to be 2 minutes. If the company wants a margin of error of no more than 0.5 minutes at a 95% confidence level ($$z_{0.025} = 1.96$$), what is the minimum sample size required?

Accepted Answer

$$n = \left( \frac{1.96 	imes 2}{0.5} \$$right)^2$$ = 61.47$$, so at least 62$

Question 16

Which of the following best explains why the Bureau of Labor Statistics (BLS) uses samples to estimate the national unemployment rate?

Accepted Answer

Because the U.S. workforce population is very large, making it impractical to survey every individual.

Question 17

Which of the following is NOT a property of the t-distribution?

Accepted Answer

It is always narrower than the normal distribution.

Question 18

A market researcher wants to estimate the proportion of consumers who prefer Brand A. In a sample of 500, 275 prefer Brand A. Calculate the 90% confidence interval for the proportion. (Use $$z_{0.05} = 1.645)$$

Accepted Answer

$$0.55 \pm (1.645 	imes \sqrt{\frac{0.55 	imes 0.45}{500}})$$, which is approximately (0.513, 0.587)

Chapter 8: Confidence Intervals – Business Statistics Study Notes

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