Q: A researcher wants to estimate the average height of high school students in a city. She collects a random sample of 100 students and calculates a sample mean of 170 cm with a sample standard deviation of 8 cm. What is the 95% confidence interval for the population mean? (Use $$z_{0.025} = 1.96)$$

$$170 \pm 1.96 \times \frac{8}{10}$$, which is $$(168.43, 171.57)$$

Q: Given a population standard deviation $$\sigma = 5$$, a desired margin of error $$d = 1$$, and a 95% confidence level ($$z_{0.025} = 1.96$$), what is the minimum sample size required?

$$n \geq \left( \frac{1.96 \times 5}{1} \$$right)^2$$ = 96.04$$

Q: A poll finds that 120 out of 400 randomly selected voters support a candidate. What is the 95% confidence interval for the proportion of supporters in the population? (Use $$z_{0.025} = 1.96)$$

$$0.3 \pm 1.96 \times \sqrt{\frac{0.3 \times 0.7}{400}} = (0.25, 0.35)$$

Q: If the sample mean is 50, the population standard deviation is 4, the sample size is 16, and the confidence level is 99% ($$z_{0.005} = 2.575$$), what is the confidence interval for the population mean?

$$50 \pm 2.575 \times \frac{4}{\sqrt{16}} = (49.04, 50.96)$$

Question 1

Which of the following best describes 'interval estimation' in statistics?

Accepted Answer

Estimating a population parameter by providing a range of values that is likely to contain the parameter, with a certain level of confidence.

Question 2

A researcher wants to estimate the average height of high school students in a city. She collects a random sample of 100 students and calculates a sample mean of 170 cm with a sample standard deviation of 8 cm. What is the 95% confidence interval for the population mean? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$170 \pm 1.96 	imes \frac{8}{10}$$, which is $$(168.43, 171.57)$$

Question 3

What does a 95% confidence interval for a population mean imply?

Accepted Answer

If we take many samples and build a confidence interval from each, about 95% of those intervals will contain the true population mean.

Question 4

Which of the following will result in a narrower confidence interval for the population mean, assuming all other factors are constant?

Accepted Answer

Decreasing the population standard deviation.

Question 5

Given a population standard deviation $$\sigma = 5$$, a desired margin of error $$d = 1$$, and a 95% confidence level ($$z_{0.025} = 1.96$$), what is the minimum sample size required?

Accepted Answer

$$n \geq \left( \frac{1.96 	imes 5}{1} \$$right)^2$$ = 96.04$$

Question 6

A poll finds that 120 out of 400 randomly selected voters support a candidate. What is the 95% confidence interval for the proportion of supporters in the population? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$0.3 \pm 1.96 	imes \sqrt{\frac{0.3 	imes 0.7}{400}} = (0.25, 0.35)$$

Question 7

Which of the following statements about the relationship between confidence level, sample size, and standard deviation is correct?

Accepted Answer

Decreasing the standard deviation decreases the width of the confidence interval.

Question 8

If the sample mean is 50, the population standard deviation is 4, the sample size is 16, and the confidence level is 99% ($$z_{0.005} = 2.575$$), what is the confidence interval for the population mean?

Accepted Answer

$$50 \pm 2.575 	imes \frac{4}{\sqrt{16}} = (49.04, 50.96)$$

Question 9

A company wants to estimate the proportion of defective products with a margin of error no greater than 0.03 at a 95% confidence level. If no prior estimate of $$p is $$available, what is the minimum sample size required? ($$z_{0.025} = 1.96)$$

Accepted Answer

$$n \geq \frac{1}{4} \left( \frac{1.96}{0.03} \$$right)^2$$ = 1067$$

Question 10

Which of the following is NOT a correct interpretation of a 99% confidence interval for a population proportion?

Accepted Answer

There is a 99% probability that the true proportion is in the calculated interval for this sample.

Question 11

A sample of 36 apples has a mean weight of 342g and a sample standard deviation of 20g. What is the 99% confidence interval for the population mean? ($$z_{0.005} = 2.575)$$

Accepted Answer

$$342 \pm 2.575 	imes \frac{20}{6} = (333.42, 350.58)$$

Question 12

Which formula is used to calculate the standard error of the sample mean when the population standard deviation is unknown and the sample size is large?

Accepted Answer

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

Question 13

In a survey, 250 out of 1000 people prefer a new product. What is the estimated standard error for the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.25 	imes 0.75}{1000}}$$

Question 14

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

Accepted Answer

Increase the sample size.

Question 15

A 95% confidence interval for the mean is calculated as (8.61, 9.79) gallons. Which of the following is the correct interpretation?

Accepted Answer

We are 95% confident that the population mean is between 8.61 and 9.79 gallons.

Statistical Inference: Estimation and Confidence Intervals

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