Q: In a study, the sample mean height of 200 adult American men is found to be 69.3 inches. The population standard deviation is known to be 2.8 inches. What is the margin of error for a 95% confidence interval? (Use $$z_{0.025} = 1.96)$$

$$E = \frac{2.8}{\sqrt{200}} \times 1.96 = 0.388$$

Question 1

Which of the following best describes a 'point estimate' in the context of estimating a population mean?

Accepted Answer

A single value calculated from a sample that is used to estimate the population mean.

Question 2

In a study, the sample mean height of 200 adult American men is found to be 69.3 inches. The population standard deviation is known to be 2.8 inches. What is the margin of error for a 95% confidence interval? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$E = \frac{2.8}{\sqrt{200}} 	imes 1.96 = 0.388$$

Question 3

What does the 'confidence level' represent in the context of a confidence interval for a population mean?

Accepted Answer

The proportion of possible samples for which the confidence interval will contain the true population mean.

Question 4

If the confidence level is 95%, what is the value of $$\alpha$$?

Accepted Answer

$$\alpha = 0.05$$

Question 5

Why do we divide $$\alpha by 2 $$when calculating the margin of error for a confidence interval?

Accepted Answer

Because the sampling error can be unusually high or unusually low, and we want to account for both tails of the normal distribution.

Question 6

Which formula correctly calculates the margin of error $$E$$ for estimating a population mean when the population standard deviation $$\sigma is $$known?

Accepted Answer

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

Question 7

A machine produces parts with a known standard deviation of 1.5 mm. A random sample of 100 parts has a mean length of 50 mm. What is the 99% confidence interval for the mean length? ($$z_{0.005} = 2.576)$$

Accepted Answer

$$(50 - 0.386, 50 + 0.386)$$

Question 8

Which of the following statements about the margin of error is TRUE?

Accepted Answer

The margin of error is always a positive value, representing the maximum likely difference between the sample mean and the population mean.

Question 9

Suppose you want to estimate the average time students spend studying per week. You select a random sample of 50 students and find a mean of 12 hours. The population standard deviation is known to be 4 hours. What is the 90% confidence interval? ($$z_{0.05} = 1.645)$$

Accepted Answer

$$(12 - 0.929, 12 + 0.929)$$

Question 10

Which of the following best explains why the confidence interval is symmetric around the sample mean?

Accepted Answer

Because the sampling distribution of the sample mean is normal and symmetric.

Question 11

A political poll reports a margin of error of 3%. What does this mean in the context of the poll?

Accepted Answer

The true population proportion is likely within 3% of the sample proportion, with the stated confidence level.

Question 12

If the sample size increases, what happens to the margin of error, assuming all other factors remain constant?

Accepted Answer

The margin of error decreases because $$\sqrt{n}$$ increases in the denominator.

Question 13

Which of the following is NOT required to calculate a confidence interval for the population mean when $$\sigma is $$known?

Accepted Answer

The population mean $$\mu$$

Question 14

A researcher wants to estimate the mean cholesterol level in a population. She knows $$\sigma = 15 mg/dL $$and takes a sample of $$n = 64$$ with $$\bar{x} = 200 mg/dL. $$What is the 95% confidence interval? ($$z_{0.025} = 1.96)$$

Accepted Answer

$$(200 - 3.675, 200 + 3.675)$$

Question 15

Which scenario would most likely require the use of a confidence interval rather than a point estimate?

Accepted Answer

When you want to express the uncertainty in your estimate of the population mean.

Question 16

If the confidence level increases from 95% to 99%, what happens to the margin of error?

Accepted Answer

The margin of error increases because the critical value $$z_{\alpha/2}$$ increases.

Question 17

A confidence interval for the mean is calculated as $$(68.91, 69.69). $$What is the sample mean and margin of error?

Accepted Answer

Sample mean $$= 69.3$$, margin of error $$= 0.39$$

Question 18

Which of the following best describes the role of the Central Limit Theorem in constructing confidence intervals for the mean?

Accepted Answer

It ensures that the sampling distribution of the sample mean is approximately normal, allowing the use of z-scores.

Confidence Intervals for One Population Mean (σ Known)

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