Q: A company wants to estimate the proportion of customers who are satisfied with their service. In a random sample of 200 customers, 150 report being satisfied. What is the 95% confidence interval for the true proportion of satisfied customers?

$$0.75 \pm 1.96 \sqrt{\frac{0.75 \times 0.25}{200}}$$

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 estimate provides a range of values likely to contain the parameter.

Question 2

Suppose you want to estimate the average delivery time for your company. You collect a random sample of 36 orders and find a sample mean of 10 hours with a known population standard deviation of 3 hours. What is the 95% confidence interval for the population mean delivery time?

Accepted Answer

$$10 \pm 1.96 \frac{3}{\sqrt{36}}$$

Question 3

Which of the following statements about unbiased estimators is TRUE?

Accepted Answer

An unbiased estimator has an expected value equal to the parameter it estimates.

Question 4

A sample of 25 students has a sample mean test score of 80 and a sample standard deviation of 10. Construct a 99% confidence interval for the population mean test score. Which formula should you use?

Accepted Answer

$$80 \pm t_{0.005,24} \frac{10}{\sqrt{25}}$$

Question 5

Which of the following actions will NOT reduce the margin of error in a confidence interval for a population mean?

Accepted Answer

Increasing the population standard deviation.

Question 6

A company wants to estimate the proportion of customers who are satisfied with their service. In a random sample of 200 customers, 150 report being satisfied. What is the 95% confidence interval for the true proportion of satisfied customers?

Accepted Answer

$$0.75 \pm 1.96 \sqrt{\frac{0.75 	imes 0.25}{200}}$$

Question 7

Which distribution should be used to construct a confidence interval for the mean when the population variance is unknown and the sample size is small?

Accepted Answer

Student's t distribution

Question 8

If a 95% confidence interval for the mean resistance of a circuit is (1.99, 2.41) ohms, what does this mean?

Accepted Answer

We are 95% confident that the true mean resistance is between 1.99 and 2.41 ohms.

Question 9

Which of the following is the correct formula for the 100(1-\alpha)% confidence interval for the variance of a normal population?

Accepted Answer

$$\left( \frac{(n-1)s^2$}{\$$chi^2$$_{1-\alpha/2}}, \frac{(n-1)s^2$}{\$$chi^2$$_{\alpha/2}} \right)$$

Question 10

A researcher wants to decrease the width of a 95% confidence interval for the mean. Which of the following strategies will achieve this?

Accepted Answer

Increase the sample size.

Question 11

Which of the following is a property of the most efficient estimator among several unbiased estimators?

Accepted Answer

It has the smallest variance among all unbiased estimators.

Question 12

If the sample size is small and the population is not normal, which of the following statements is correct regarding confidence intervals for the variance?

Accepted Answer

You cannot apply the methods for confidence intervals for the variance.

Question 13

A sample of 100 batteries has a sample mean life of 500 hours and a known population standard deviation of 50 hours. What is the margin of error for a 90% confidence interval for the mean battery life?

Accepted Answer

$$1.645 	imes \frac{50}{\sqrt{100}}$$

Question 14

Which of the following best describes the interpretation of a 95% confidence level?

Accepted Answer

In repeated sampling, 95% of all confidence intervals constructed will contain the true parameter.

Question 15

A random sample of 50 light bulbs has a sample mean lifetime of 1200 hours and a sample standard deviation of 100 hours. What is the 95% confidence interval for the mean lifetime?

Accepted Answer

$$1200 \pm t_{0.025,49} \frac{100}{\sqrt{50}}$$

Question 16

Which of the following is NOT an assumption for constructing a confidence interval for the mean using the Z distribution?

Accepted Answer

The population mean is unknown but the population variance is also unknown.

Estimation: Single Population – Confidence Intervals and Point Estimation

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