Q: A company wants to estimate the average number of hours employees spend in meetings per week. They sample 25 employees and find a sample mean of 6 hours with a sample standard deviation of 1.5 hours. What is the 95% confidence interval for the population mean? (Assume the population is normally distributed.)

$$6 \pm 2.064 \times \frac{1.5}{\sqrt{25}}$$

Q: A hotel chain wants to estimate the proportion of guests who use their gym facilities. In a random sample of 200 guests, 60 used the gym. What is the 95% confidence interval for the true proportion of guests who use the gym?

$$0.3 \pm 1.96 \times \sqrt{\frac{0.3 \times 0.7}{200}}$$

Q: A company samples 100 customers and finds that 62 used a promotional voucher. What is the standard error for the sample proportion?

$$\sqrt{\frac{0.62 \times 0.38}{100}}$$

Question 1

Which of the following best describes the difference between a point estimate and a confidence interval estimate?

Accepted Answer

A point estimate provides a single value as an estimate of a population parameter, while a confidence interval provides a range of values that is likely to contain the population parameter.

Question 2

A company wants to estimate the average number of hours employees spend in meetings per week. They sample 25 employees and find a sample mean of 6 hours with a sample standard deviation of 1.5 hours. What is the 95% confidence interval for the population mean? (Assume the population is normally distributed.)

Accepted Answer

$$6 \pm 2.064 	imes \frac{1.5}{\sqrt{25}}$$

Question 3

Which of the following actions will decrease the margin of error in a confidence interval for a population mean, assuming all other factors remain constant?

Accepted Answer

Increase the sample size.

Question 4

A hotel chain wants to estimate the proportion of guests who use their gym facilities. In a random sample of 200 guests, 60 used the gym. What is the 95% confidence interval for the true proportion of guests who use the gym?

Accepted Answer

$$0.3 \pm 1.96 	imes \sqrt{\frac{0.3 	imes 0.7}{200}}$$

Question 5

Which of the following is NOT a step in constructing a confidence interval for a population mean?

Accepted Answer

Set the null and alternative hypotheses.

Question 6

A business wants to test if the average delivery time is less than 3 days. Which of the following correctly states the null and alternative hypotheses?

Accepted Answer

$$H_0$$: \mu = 3$$, H_a$$: \mu \u003c 3$$

Question 7

In hypothesis testing, what does a Type I error represent?

Accepted Answer

Rejecting the null hypothesis when it is true.

Question 8

A manufacturer claims their light bulbs last at least 1,000 hours. A sample of 36 bulbs has a mean life of 980 hours and a known population standard deviation of 60 hours. At $$\alpha = 0.05$$, what is the critical z-value for this one-tailed test?

Accepted Answer

$$1.645$$

Question 9

Which formula is used to calculate the test statistic for a hypothesis test for a population mean when the population standard deviation is known?

Accepted Answer

$$z = \frac{\bar{x} - \mu_0$}{\sigma / \sqrt{n}}$$

Question 10

A restaurant wants to know if more than 20% of its customers order dessert. In a sample of 150 customers, 36 ordered dessert. What is the value of the sample proportion?

Accepted Answer

$$0.24$$

Question 11

When should you use the t-distribution instead of the z-distribution for constructing a confidence interval for the mean?

Accepted Answer

When the population standard deviation is unknown and the sample size is small.

Question 12

A company samples 100 customers and finds that 62 used a promotional voucher. What is the standard error for the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.62 	imes 0.38}{100}}$$

Question 13

If a 95% confidence interval for a mean is (10, 20), which of the following statements is correct?

Accepted Answer

We are 95% confident that the population mean lies between 10 and 20.

Question 14

A business wants to determine the minimum sample size needed to estimate a population mean with a margin of error of 2 units at 95% confidence. The population standard deviation is estimated to be 10 units. Which equation should be used?

Accepted Answer

$$n = \left( \frac{z_{0.025} 	imes 10}{2} \$$right)^2$$

Question 15

In a hypothesis test for a population proportion, which of the following is the correct formula for the test statistic?

Accepted Answer

$$z = \frac{\hat{p} - p_0$}{\sqrt{\frac{p_0$ (1 - p_0$)}{n}}}$$

Question 16

A company is considering increasing prices if more than 10% of customers would cancel their orders. In a sample of 100 customers, 14 said they would cancel. At $$\alpha = 0.05$$, what is the null hypothesis?

Accepted Answer

$$H_0$$: p = 0.10$$

Question 17

Which of the following best describes the Central Limit Theorem?

Accepted Answer

For large sample sizes, the sampling distribution of the sample mean is approximately normal, regardless of the population distribution.

Question 18

A maintenance company advertises average service times as 40 minutes. A sample of 100 services has a mean of 43 minutes and a known population standard deviation of 10 minutes. What is the value of the test statistic for testing if the mean service time exceeds 40 minutes?

Accepted Answer

$$z = \frac{43 - 40}{10 / \sqrt{100}} = 3$$

Question 19

If you want to be more confident that your confidence interval contains the true population parameter, what should you do?

Accepted Answer

Increase the confidence level, which will widen the interval.

Statistical Inference and Confidence Intervals in Business Statistics

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