Q: A researcher wants a 95% confidence interval for the mean diameter of logs with a population standard deviation of 4.8 inches. If the sample mean is 12.09 inches and the sample size is 100, what is the margin of error?

$$1.96 \times \frac{4.8}{\sqrt{100}}$$

Q: Suppose a sample of 40 cans from a population with $$\sigma = 0.20$$ produced a sample mean equal to 12.09. What is the 95% confidence interval for the mean?

$$12.09 \pm 1.96 \times \frac{0.20}{\sqrt{40}}$$

Question 1

Which of the following best describes a point estimate in statistics?

Accepted Answer

A value calculated from sample data used to estimate a population parameter.

Question 2

Suppose you want to estimate the mean age of people in a city. Which of the following is the best approach if the population is very large?

Accepted Answer

Take a sufficiently large random sample and use the sample mean as an estimate.

Question 3

A confidence interval for the population mean $$\mu is $$constructed as $$\bar{x} \pm z \frac{\sigma}{\sqrt{n}}. $$What does $$z$$ represent in this formula?

Accepted Answer

The critical value from the standard normal distribution for the desired confidence level.

Question 4

If you increase the sample size $$n$$ while keeping all other factors constant, what happens to the width of the confidence interval for the mean?

Accepted Answer

The interval becomes narrower.

Question 5

A researcher wants a 95% confidence interval for the mean diameter of logs with a population standard deviation of 4.8 inches. If the sample mean is 12.09 inches and the sample size is 100, what is the margin of error?

Accepted Answer

$$1.96 	imes \frac{4.8}{\sqrt{100}}$$

Question 6

Which of the following statements about confidence intervals is TRUE?

Accepted Answer

A 95% confidence interval means that if we repeated the sampling process many times, about 95% of the intervals would contain the true population mean.

Question 7

What is the effect of increasing the confidence level from 95% to 99% on the confidence interval for a population mean?

Accepted Answer

The interval becomes wider.

Question 8

Suppose a sample of 40 cans from a population with $$\sigma = 0.20$$ produced a sample mean equal to 12.09. What is the 95% confidence interval for the mean?

Accepted Answer

$$12.09 \pm 1.96 	imes \frac{0.20}{\sqrt{40}}$$

Question 9

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

Accepted Answer

The population mean is known.

Question 10

A statistic is called unbiased if:

Accepted Answer

Its value in repeated samples is equal to the true value of the parameter.

Question 11

If a calculation for required sample size yields 58.2, what should you do?

Accepted Answer

Round up to 59 observations.

Question 12

Which formula would you use to find the minimum sample size $$n$$ for a desired margin of error $$e in $$estimating a population mean with known standard deviation $$\sigma$$ and critical value $$z$$?

Accepted Answer

$$n = \left( \frac{z \sigma}{e} \$$right)^2$$

Question 13

Which of the following best interprets the meaning of a random interval in the context of confidence intervals?

Accepted Answer

The interval varies from sample to sample, but the center is not random.

Question 14

Why is it important to use a simple random sample (SRS) when estimating population parameters?

Accepted Answer

It ensures that every member of the population has an equal chance of being selected, reducing bias.

Question 15

Which of the following would result in a wider confidence interval for the mean diameter of logs?

Accepted Answer

Decreasing the sample size.

Question 16

A manager wants to estimate the mean diameter of logs within 0.50 inches at 90% confidence, with a population standard deviation of 4.8 inches. Which formula should be used to determine the required sample size?

Accepted Answer

$$n = \left( \frac{z 	imes 4.8}{0.50} \$$right)^2$$

Question 17

Which of the following is a plausible consequence of using a biased estimator?

Accepted Answer

The sample statistic consistently over- or underestimates the population parameter.

Question 18

If the population standard deviation $$\sigma is $$unknown, which of the following should be used to construct a confidence interval for the mean?

Accepted Answer

Use the sample standard deviation $$s$$ and the $$t-$$distribution.

Question 19

Which of the following is NOT a way to reduce the margin of error in a confidence interval?

Accepted Answer

Decrease the sample size.

Point Estimation and Confidence Intervals in Statistics

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