Q: A researcher wants to estimate the average height of high school students in a city. She takes a random sample of 50 students and calculates a sample mean of 165 cm with a sample standard deviation of 8 cm. What is the 95% confidence interval for the population mean?

$$165 \pm 1.96 \times \frac{8}{\sqrt{50}} = (162.78, 167.22)$$

Q: Which formula is used to calculate the margin of error for the confidence interval of a population mean when the population standard deviation $$\sigma is $$known?

$$z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$$

Q: A factory produces sugar bags with weights that follow a normal distribution with a known standard deviation of 1 ounce. A random sample of 25 bags has a mean weight of 49.8 ounces. What is the 95% confidence interval for the mean weight?

$$(49.8 \pm 1.96 \times \frac{1}{\sqrt{25}}) = (49.408, 50.192)$$

Q: A survey of 500 people finds that 41 are unemployed. What is the 95% confidence interval for the unemployment rate in the population?

$$0.082 \pm 1.96 \times \sqrt{\frac{0.082 \times 0.918}{500}} = (0.058, 0.106)$$

Question 1

Which of the following best describes the concept of 'interval estimation' in statistical inference?

Accepted Answer

Estimating a population parameter by providing a range of values that is likely to contain the parameter, with a specified probability.

Question 2

A researcher wants to estimate the average height of high school students in a city. She takes a random sample of 50 students and calculates a sample mean of 165 cm with a sample standard deviation of 8 cm. What is the 95% confidence interval for the population mean?

Accepted Answer

$$165 \pm 1.96 	imes \frac{8}{\sqrt{50}} = (162.78, 167.22)$$

Question 3

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

Accepted Answer

If we repeat the sampling process many times, about 95% of the calculated confidence intervals will contain the true population mean.

Question 4

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

Accepted Answer

$$z_{\alpha/2} 	imes \frac{\sigma}{\sqrt{n}}$$

Question 5

If the confidence level is increased from 95% to 99%, what happens to the width of the confidence interval?

Accepted Answer

The width of the confidence interval increases.

Question 6

A factory produces sugar bags with weights that follow a normal distribution with a known standard deviation of 1 ounce. A random sample of 25 bags has a mean weight of 49.8 ounces. What is the 95% confidence interval for the mean weight?

Accepted Answer

$$(49.8 \pm 1.96 	imes \frac{1}{\sqrt{25}}) = (49.408, 50.192)$$

Question 7

Which of the following factors will decrease the width of a confidence interval for a population mean?

Accepted Answer

Decreasing the confidence level.

Question 8

In the context of confidence intervals, what does the significance level $$\alpha$$ represent?

Accepted Answer

The probability that the confidence interval does not contain the population parameter.

Question 9

A survey of 500 people finds that 41 are unemployed. What is the 95% confidence interval for the unemployment rate in the population?

Accepted Answer

$$0.082 \pm 1.96 	imes \sqrt{\frac{0.082 	imes 0.918}{500}} = (0.058, 0.106)$$

Question 10

Which formula is used to determine the minimum sample size required to estimate a population mean with a margin of error $$d$$ and confidence level $$1-\alpha$$?

Accepted Answer

$$n = \left( \frac{z_{\alpha/2} \cdot \sigma}{d} \$$right)^2$$

Question 11

If the population proportion $$p is $$unknown, which value should be used for $$p(1-p) to $$ensure the minimum sample size for a given margin of error and confidence level?

Accepted Answer

$$p(1-p)$$ should be set to $$\frac{1}{4}$$

Question 12

A company wants to estimate the proportion of customers who purchase a premium product. In a sample of 250 customers, 70 purchased the premium product. What is the estimated standard error for the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.28 	imes 0.72}{250}} = 0.028$$

Question 13

Which of the following statements about confidence intervals is correct?

Accepted Answer

The confidence interval is wider for higher confidence levels.

Question 14

In a poll, 250 customers are surveyed and 100 have spent more than 100,000 won. What is the 99% confidence interval for the proportion of customers spending more than 100,000 won?

Accepted Answer

$$0.4 \pm 2.58 	imes \sqrt{\frac{0.4 	imes 0.6}{250}} = (0.32, 0.48)$$

Question 15

Which of the following is NOT a correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

There is a 95% probability that the sample mean is the true mean.

Question 16

Which of the following will NOT affect the width of a confidence interval for a population mean?

Accepted Answer

Sample mean

Question 17

A researcher wants to estimate the average monthly income of households in Seoul. The range of incomes is from 1,000,000 to 2,600,000 won. If the desired margin of error is 50,000 won and the confidence level is 95%, what is the minimum sample size required?

Accepted Answer

$$n = \left[ \frac{1.96 	imes (1600/4)}{50} ight]^2 = 246$$

Question 18

Which of the following is the correct formula for the confidence interval of a population proportion $$p$$ when the sample size is large?

Accepted Answer

$$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Statistical Inference: Estimation and Confidence Intervals

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