Q: A researcher wants to estimate the mean weight of apples in an orchard. She collects a sample of 25 apples with a sample mean of 150g and a sample standard deviation of 10g. What is the 95% confidence interval for the population mean? (Use $$t_{0.025, 24} \approx 2.064)$$

$$150 \pm 2.064 \times \dfrac{10}{\sqrt{25}}$$

Q: A medical researcher wants to estimate the average recovery time for a new drug. She collects a sample of 16 patients, finds a sample mean of 8 days, and a sample standard deviation of 2 days. What is the 90% confidence interval for the mean recovery time? (Use $$t_{0.05, 15} \approx 1.753)$$

$$8 \pm 1.753 \times \dfrac{2}{\sqrt{16}}$$

Q: A pollster wants to estimate the proportion of voters who support a candidate with a margin of error of 0.03 and a 95% confidence level. If no prior estimate is available, what formula should be used to determine the minimum sample size?

$$n = \dfrac{(1.96)^2$ \times 0.25}{(0.03)^2$}$$

Question 1

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

Accepted Answer

A single value used to approximate a population parameter.

Question 2

A survey of 1000 adults finds that 60% support a new policy. What is the best point estimate for the population proportion $$p$$?

Accepted Answer

$$p = 0.60$$

Question 3

What is the formula for the margin of error $$E$$ when estimating a population proportion $$p$$ with known sample proportion $$\hat{p}$$?

Accepted Answer

$$E = z_{\alpha/2} \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$$

Question 4

If you want a 95% confidence interval for a population mean and the population standard deviation $$\sigma is $$unknown, which distribution should you use?

Accepted Answer

Student $$t$$ distribution

Question 5

Which of the following statements about the Student $$t$$ distribution is TRUE?

Accepted Answer

It has greater variability than the standard normal distribution for small sample sizes.

Question 6

A researcher wants to estimate the mean weight of apples in an orchard. She collects a sample of 25 apples with a sample mean of 150g and a sample standard deviation of 10g. What is the 95% confidence interval for the population mean? (Use $$t_{0.025, 24} \approx 2.064)$$

Accepted Answer

$$150 \pm 2.064 	imes \dfrac{10}{\sqrt{25}}$$

Question 7

Which of the following is the correct formula for the minimum sample size $$n$$ needed to estimate a population proportion $$p$$ with margin of error $$E$$ and no prior estimate for $$\hat{p}$$?

Accepted Answer

$$n = \dfrac{z_{\alpha/2}^2 \cdot 0.25}{E^2$}$$

Question 8

If the computed sample size for a study is 23.4, what should the researcher use as the final sample size?

Accepted Answer

24

Question 9

Which of the following best explains why the Student $$t$$ distribution is wider than the standard normal distribution for small sample sizes?

Accepted Answer

There is greater variability in sample estimates when the sample size is small.

Question 10

A 95% confidence interval for a population proportion $$p is $$calculated as $$(0.45, 0.55). $$What does this mean?

Accepted Answer

We are 95% confident that the true population proportion is between 0.45 and 0.55.

Question 11

Which critical value $$z_{\alpha/2}$$ would you use for a 99% confidence interval?

Accepted Answer

$$2.576$$

Question 12

A medical researcher wants to estimate the average recovery time for a new drug. She collects a sample of 16 patients, finds a sample mean of 8 days, and a sample standard deviation of 2 days. What is the 90% confidence interval for the mean recovery time? (Use $$t_{0.05, 15} \approx 1.753)$$

Accepted Answer

$$8 \pm 1.753 	imes \dfrac{2}{\sqrt{16}}$$

Question 13

Which of the following is NOT a property of the Student $$t$$ distribution?

Accepted Answer

It has a standard deviation less than 1 for all sample sizes.

Question 14

If $$\alpha = 0.10$$, what is the corresponding confidence level?

Accepted Answer

90%

Question 15

A pollster wants to estimate the proportion of voters who support a candidate with a margin of error of 0.03 and a 95% confidence level. If no prior estimate is available, what formula should be used to determine the minimum sample size?

Accepted Answer

$$n = \dfrac{(1.96)^2$ 	imes 0.25}{(0.03)^2$}$$

Question 16

Which scenario requires the use of the $$t$$ distribution rather than the $$z$$ distribution?

Accepted Answer

Estimating a population mean when the population standard deviation is unknown and the sample size is small.

Chapter 7: Estimating Population Parameters and Sample Sizes

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