Q: A random sample of 70 USC students finds that 60% played a sport in high school. What is the margin of error for a 95% confidence interval?

$$1.96 \times \sqrt{\frac{0.60 \times 0.40}{70}}$$

Q: A sample of 1,555 people finds a mean of 2.94 hours of TV watched per day, with a standard deviation of 2.84. What is the 95% confidence interval for the population mean?

$$2.94 \pm 1.96 \times \frac{2.84}{\sqrt{1555}}$$

Q: What is the formula for the Agresti-Coull adjusted sample proportion $$\tilde{p}$$?

$$\tilde{p} = \frac{x + 2}{n + 4}$$

Q: A sample of 20 USC students has a mean Amazon spending of $$55.50 and a sample standard deviation of 29.06. $$What is the 95% confidence interval for the true mean, using the t-distribution?

$$55.50 \pm 2.093 \times \frac{29.06}{\sqrt{20}}$$

Question 1

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

Accepted Answer

A single value calculated from a sample that is our best guess for an unknown population parameter.

Question 2

A pharmaceutical company tests a new drug on 50 patients, and 22 report better heart health. Construct a 95% confidence interval for the true proportion of patients who will see improvement. Which formula should you use?

Accepted Answer

Use the Wald interval: $$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 3

What happens to the width of a confidence interval as the sample size $$n$$ increases?

Accepted Answer

The interval becomes narrower.

Question 4

Suppose you want to construct a 99% confidence interval for a population proportion. What critical value should you use from the z-table?

Accepted Answer

$$2.575$$

Question 5

A random sample of 70 USC students finds that 60% played a sport in high school. What is the margin of error for a 95% confidence interval?

Accepted Answer

$$1.96 	imes \sqrt{\frac{0.60 	imes 0.40}{70}}$$

Question 6

Why are 100% confidence intervals not constructed in practice?

Accepted Answer

Because the interval would be so wide as to be useless for inference.

Question 7

Which of the following is a key assumption for constructing a confidence interval for a proportion?

Accepted Answer

$$n \hat{p} \geq 15$$ and $$n(1-\hat{p}) \geq 15$$

Question 8

If you increase the confidence level from 95% to 99%, what happens to the confidence interval?

Accepted Answer

The interval becomes wider.

Question 9

A sample of 1,555 people finds a mean of 2.94 hours of TV watched per day, with a standard deviation of 2.84. What is the 95% confidence interval for the population mean?

Accepted Answer

$$2.94 \pm 1.96 	imes \frac{2.84}{\sqrt{1555}}$$

Question 10

Which method is recommended for constructing confidence intervals for proportions when sample sizes are small?

Accepted Answer

Agresti-Coull interval

Question 11

What is the formula for the Agresti-Coull adjusted sample proportion $$	ilde{p}$$?

Accepted Answer

$$	ilde{p} = \frac{x + 2}{n + 4}$$

Question 12

When constructing a confidence interval for a population mean with a small sample size ($$n \u003c 30$$), which distribution should be used?

Accepted Answer

t-distribution with $$df = n - 1$$

Question 13

Why is the degrees of freedom for a t-distribution $$n - 1$$ when estimating a mean?

Accepted Answer

Because one value is fixed by knowing the sample mean, leaving $$n - 1$$ values free to vary.

Question 14

A sample of 20 USC students has a mean Amazon spending of $$55.50 and a sample standard deviation of 29.06. $$What is the 95% confidence interval for the true mean, using the t-distribution?

Accepted Answer

$$55.50 \pm 2.093 	imes \frac{29.06}{\sqrt{20}}$$

Question 15

Which of the following is NOT a valid interpretation of a 95% confidence interval for a population mean?

Accepted Answer

There is a 95% probability that the true mean is in the interval calculated from our sample.

Question 16

Which of the following is a consequence of using a convenience sample instead of a random sample when constructing a confidence interval?

Accepted Answer

The results may not be generalizable to the entire population.

Question 17

A Sweet Cream Company surveys 20 customers and 18 say they would purchase a new ice cream flavor. Using the Agresti-Coull method, what is the adjusted sample size $$	ilde{n}$$?

Accepted Answer

$$	ilde{n} = 24$$

Question 18

Which of the following statements about post hoc analysis is correct?

Accepted Answer

It is generally discouraged because it can lead to biased conclusions.

Confidence Intervals and Hypothesis Testing: Study Notes for Statistics for Business

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