Q: A researcher collects a sample of $$n = 15$$ birth weights with a sample mean of $$30.9 kg $$and a sample standard deviation of $$2.9 kg. $$What is the correct formula for the margin of error $$E$$ for a 95% confidence interval for the mean?

$$E = t_{\alpha/2} \times \dfrac{s}{\sqrt{n}}$$

Question 1

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

Accepted Answer

A single value computed from sample data to estimate a population parameter.

Question 2

When constructing a confidence interval for a population mean with an unknown population standard deviation, which distribution should be used?

Accepted Answer

The t distribution, which depends on sample size.

Question 3

A researcher collects a sample of $$n = 15$$ birth weights with a sample mean of $$30.9 kg $$and a sample standard deviation of $$2.9 kg. $$What is the correct formula for the margin of error $$E$$ for a 95% confidence interval for the mean?

Accepted Answer

$$E = t_{\alpha/2} 	imes \dfrac{s}{\sqrt{n}}$$

Question 4

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 5

Given a sample mean $$\bar{x} = 30.9$$, margin of error $$E = 0.85$$, and a 95% confidence level, what is the correct confidence interval for the population mean $$\mu$$?

Accepted Answer

$$(30.9 - 0.85, 30.9 + 0.85)$$

Question 6

If the confidence level is increased from 95% to 99%, what happens to the width of the confidence interval for the mean, assuming all other factors remain constant?

Accepted Answer

The interval becomes wider.

Question 7

Which formula is used to construct a confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

$$\bar{x} \pm t_{\alpha/2} \dfrac{s}{\sqrt{n}}$$

Question 8

A sample of $$n = 40$$ adults in a weight loss program has a sample mean weight loss of $$8.2 kg $$and a sample standard deviation of $$4.8 kg. $$What is the correct formula to calculate the 90% confidence interval for the mean weight loss?

Accepted Answer

$$\bar{x} \pm t_{\alpha/2} \dfrac{s}{\sqrt{n}}$$

Question 9

Which of the following is NOT a step in constructing a confidence interval for a population mean?

Accepted Answer

Multiply the sample mean by the sample size.

Question 10

Why is the t distribution used instead of the z distribution when estimating a population mean with an unknown standard deviation?

Accepted Answer

Because the t distribution accounts for the additional uncertainty from estimating the standard deviation.

Question 11

A 95% confidence interval for a mean is calculated as $$(27.0, 33.0). $$What is the point estimate of the mean?

Accepted Answer

$$30.0$$

Question 12

Which of the following best describes the relationship between confidence level and alpha ($$\alpha$$)?

Accepted Answer

Confidence level $$= 1 - \alpha$$

Question 13

For a sample size $$n = 23$$ and a 98% confidence level, which value should be used to find the critical value $$t_{\alpha/2}$$?

Accepted Answer

Degrees of freedom $$= 22$$

Question 14

Which of the following is TRUE about the margin of error in a confidence interval for the mean?

Accepted Answer

It decreases as the sample size increases.

Question 15

A medical researcher wants to estimate the average cholesterol level in adults. She collects a sample of $$n = 100$$ adults and calculates a sample mean of $$200 mg/dL $$and a sample standard deviation of $$30 mg/dL. $$What is the formula for the 80% confidence interval for the mean cholesterol level?

Accepted Answer

$$200 \pm t_{\alpha/2} \dfrac{30}{\sqrt{100}}$$

Question 16

Which of the following is the correct interpretation of a 90% confidence interval for a population mean?

Accepted Answer

If we repeated the sampling process many times, 90% of the constructed intervals would contain the true population mean.

Question 17

When constructing a confidence interval for a mean, which of the following factors will NOT affect the width of the interval?

Accepted Answer

The color of the graph used to display the interval

Question 18

A sample of $$n = 12$$ has a 95% confidence level. Which table should be used to find the critical value for constructing the confidence interval for the mean?

Accepted Answer

The t table (degrees of freedom $$= 11)$$

Question 19

Which of the following is a reason why the 95% confidence level is commonly used in practice?

Accepted Answer

It provides a good balance between precision and reliability.

Estimating a Population Mean: Confidence Intervals and Levels

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