Question 1

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

Accepted Answer

$$E = z_c \cdot \frac{\sigma}{\sqrt{n}}$$

Question 2

A sample of $$n=34$$ students at THS has a sample mean of $$\bar{x} = 8.66$$ hours of sleep. Is this value exactly equal to the population mean $$\mu$$? Why or why not?

Accepted Answer

No, $$\bar{x} is $$only an estimate of $$\mu$$ based on the sample.

Question 3

Given $$\sigma = 1.27$$, $$n = 34$$, and a 90% confidence level ($$z_c = 1.645$$), what is the margin of error $$E$$?

Accepted Answer

$$E = 1.645 \cdot \frac{1.27}{\sqrt{34}}$$

Question 4

If a 90% confidence interval for the mean hours of sleep is $$(8.30, 9.02)$$, how should you interpret this interval?

Accepted Answer

We are 90% confident that the true mean hours of sleep for all THS students is between 8.30 and 9.02 hours.

Question 5

Why is it appropriate to use $$z-$$scores for constructing a confidence interval in this scenario, even though the population distribution is unknown?

Accepted Answer

Because the sample size is large ($$n \geq 30$$), the Central Limit Theorem applies.

Question 6

What is the minimum sample size $$n$$ needed to estimate the population mean within $$E = 0.5$$ hours with 95% confidence, given $$\sigma = 1.27$$ and $$z_{0.95} = 1.96$$?

Accepted Answer

$$n = \left( \frac{1.96 \cdot 1.27}{0.5} \$$right)^2$$

Question 7

If the calculated confidence interval is $$(8.30, 9.02)$$ and the true population mean is $$8.33$$, did the interval capture the true mean?

Accepted Answer

Yes, because $$8.33 is $$within the interval.

Question 8

Which of the following best describes the meaning of a 90% confidence level?

Accepted Answer

If we took many samples and built a confidence interval from each, about 90% of those intervals would contain the true population mean.

Question 9

Suppose you want to decrease the margin of error in your confidence interval. Which of the following changes would achieve this?

Accepted Answer

Increase the sample size $$n.$$

Question 10

A researcher wants to estimate the average number of hours students sleep with a margin of error of $$0.25$$ hours at a 99% confidence level ($$z_{0.99} = 2.575) $$and $$\sigma = 1.27. $$What is the minimum sample size required?

Accepted Answer

$$n = \left( \frac{2.575 \cdot 1.27}{0.25} \$$right)^2$$

Question 11

If the sample mean $$\bar{x} is$$ $$8.66$$ and the margin of error $$E is$$ $$0.36$$, what is the 90% confidence interval for the population mean?

Accepted Answer

$$(8.30, 9.02)$$

Question 12

Which of the following is NOT a requirement for constructing a confidence interval for the mean using $$z-$$scores?

Accepted Answer

The population mean $$\mu$$ must be known.

Constructing and Interpreting Confidence Intervals for the Population Mean (σ Known)

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