Q: What is the formula for the point estimate of the mean from a confidence interval with upper and lower limits?

$$\bar{x} = \frac{\text{upper confidence limit} + \text{lower confidence limit}}{2}$$

Question 1

A researcher wants to estimate the mean age of STLCC students in Fall 2024. The sample mean is 24.8 years with a standard deviation of 9.8 years and a sample size of 15,649. Which formula should be used to calculate the 95% confidence interval for the population mean when the population standard deviation is not known?

Accepted Answer

$$\bar{x} \pm t \frac{s}{\sqrt{n}}$$

Question 2

Given a confidence interval of (24.6, 25.0) for the mean age of STLCC students, what is the margin of error $$E$$?

Accepted Answer

$$E = \frac{25.0 - 24.6}{2} = 0.2$$

Question 3

Which of the following statements best describes the interpretation of a 95% confidence interval for the mean age of STLCC students?

Accepted Answer

We are 95% confident that the true mean of the population lies within the calculated interval.

Question 4

If the sample size increases while keeping the confidence level and sample standard deviation constant, what happens to the width of the confidence interval?

Accepted Answer

The width of the confidence interval decreases.

Question 5

Which method should be used to estimate a population mean when the population is not normally distributed and the sample size is less than or equal to 30?

Accepted Answer

Use the bootstrapping or a nonparametric method.

Question 6

What is the formula for the point estimate of the mean from a confidence interval with upper and lower limits?

Accepted Answer

$$\bar{x} = \frac{	ext{upper confidence limit} + 	ext{lower confidence limit}}{2}$$

Question 7

A sample of ages has a minimum of 16 and a maximum of 70. Using the range rule of thumb, what is the estimated standard deviation $$\sigma$$?

Accepted Answer

$$\sigma = \frac{70 - 16}{4} = 13.5$$

Question 8

Which of the following is a requirement for finding the sample size needed to estimate a population mean?

Accepted Answer

The sample must be a simple random sample.

Question 9

Why is it recommended to use a larger estimate for $$\sigma$$ when determining sample size if the true value is unknown?

Accepted Answer

A larger $$\sigma$$ will ensure the sample size is not too small, making the estimate conservative.

Question 10

Which statement about the Student $$t$$ distribution is correct?

Accepted Answer

The Student $$t$$ distribution has a mean of $$t = 0$$ and its standard deviation is greater than 1.

Question 11

A medical researcher wants to estimate the mean recovery time for a new drug. The sample mean is 8 days, sample standard deviation is 2 days, and sample size is 25. Which distribution should be used to construct the confidence interval?

Accepted Answer

Student $$t$$ distribution

Question 12

If you want a narrower confidence interval for the mean, which of the following changes will achieve this?

Accepted Answer

Decrease the standard deviation.

Question 13

Which formula is used to calculate the margin of error $$E$$ from a confidence interval with upper and lower limits?

Accepted Answer

$$E = \frac{	ext{upper confidence limit} - 	ext{lower confidence limit}}{2}$$

Question 14

A sample of 81 STLCC students is used to estimate the mean age. What are the degrees of freedom for the Student $$t$$ distribution?

Accepted Answer

$$df = 80$$

Question 15

Which confidence interval will be the widest for the same sample data: 90%, 95%, or 99%?

Accepted Answer

99% confidence interval

Question 16

A city planner wants to estimate the mean commute time for residents. She collects a simple random sample of 50 commute times, finds the sample mean and standard deviation, and the population is normally distributed. Which method should she use to construct the confidence interval?

Accepted Answer

Student $$t$$ distribution

Question 17

Which of the following is NOT a key concept in estimating a population mean?

Accepted Answer

Population proportion

Question 18

A researcher uses prior results to estimate $$\sigma$$ for a new study. Why might she choose a value for $$\sigma$$ that is larger than the true value?

Accepted Answer

To ensure the sample size is not too small and the estimate is conservative.

Question 19

Which of the following best describes the relationship between the Student $$t$$ distribution and the standard normal distribution as sample size increases?

Accepted Answer

The Student $$t$$ distribution approaches the standard normal distribution.

Estimating Parameters and Determining Sample Sizes (Chapter 7.2 Study Notes)

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