Q: A researcher wants to estimate the proportion of voters who support a new policy. She collects a random sample and calculates $$\hat{p} = 0.40$$ with a standard error $$se = 0.03. $$What is the 95% confidence interval for the population proportion?

$$0.40 \pm 1.96 \times 0.03 = (0.34, 0.46)$$

Q: A pollster wants a margin of error of $$m = 0.02$$ for a 95% confidence interval for a proportion, and expects $$p = 0.5. $$What sample size $$n is $$required?

$$n = \dfrac{0.5 \times 0.5 \times (1.96)^2$}{(0.02)^2$}$$

Question 1

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

Accepted Answer

A point estimate is a single number that is our best guess for the parameter.

Question 2

Which property makes the sample mean $$\bar{x} a $$good estimator of the population mean $$\mu$$?

Accepted Answer

It is an unbiased estimator, meaning its sampling distribution is centered at $$\mu.$$

Question 3

A researcher wants to estimate the proportion of voters who support a new policy. She collects a random sample and calculates $$\hat{p} = 0.40$$ with a standard error $$se = 0.03. $$What is the 95% confidence interval for the population proportion?

Accepted Answer

$$0.40 \pm 1.96 	imes 0.03 = (0.34, 0.46)$$

Question 4

Why is an interval estimate generally preferred over a point estimate when reporting results from a statistical study?

Accepted Answer

An interval estimate incorporates a margin of error, providing information about the accuracy of the estimate.

Question 5

Suppose you want to construct a 99% confidence interval for a population proportion. Which z-score should you use?

Accepted Answer

$$z = 2.58$$

Question 6

A clinical trial tests a new vaccine. Out of $$n = 3900$$ subjects, $$26$$ develop the flu. Calculate the standard error for the sample proportion.

Accepted Answer

$$se = \sqrt{\dfrac{0.0067(1-0.0067)}{3900}} \approx 0.0013$$

Question 7

Which of the following best describes the meaning of a 95% confidence interval for a population proportion?

Accepted Answer

If we repeated the sampling process many times, about 95% of the intervals constructed would contain the true population proportion.

Question 8

A pollster wants a margin of error of $$m = 0.02$$ for a 95% confidence interval for a proportion, and expects $$p = 0.5. $$What sample size $$n is $$required?

Accepted Answer

$$n = \dfrac{0.5 	imes 0.5 	imes (1.96)^2$}{(0.02)^2$}$$

Question 9

Which of the following is a necessary condition for the validity of the normal approximation in constructing a confidence interval for a proportion?

Accepted Answer

Both $$np \geq 15$$ and $$n(1-p) \geq 15$$

Question 10

A sample of $$n = 11$$ eBay auctions for iPhones yields a sample mean closing price of $$\bar{x} = 583.64$$ and sample standard deviation $$s = 22.15. $$What is the standard error of the mean?

Accepted Answer

$$se = \dfrac{22.15}{\sqrt{11}} \approx 6.68$$

Question 11

When constructing a confidence interval for a population mean and the population standard deviation is unknown, which distribution should be used?

Accepted Answer

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

Question 12

Which of the following factors will decrease the margin of error for a confidence interval?

Accepted Answer

Increasing the sample size

Question 13

A researcher wants to estimate the mean number of years of education in a population. The data range from 0 to 18 years. What is a reasonable estimate for the population standard deviation $$\sigma if no $$prior information is available?

Accepted Answer

$$\sigma \approx \dfrac{18 - 0}{6} = 3$$

Question 14

If a sample does not have at least 15 successes and 15 failures, what adjustment should be made to the confidence interval for a proportion?

Accepted Answer

Add 2 to the number of successes and 2 to the number of failures, and add 4 to the sample size.

Question 15

Which of the following statements about the t distribution is correct?

Accepted Answer

It is bell-shaped and symmetric about 0, with thicker tails than the standard normal distribution.

Question 16

A 99% confidence interval for a population proportion is wider than a 95% confidence interval because:

Accepted Answer

The z-score for 99% confidence is larger, increasing the margin of error.

Question 17

Which of the following is NOT a factor that affects the choice of sample size for a study?

Accepted Answer

The color of the data points

Question 18

In practice, when constructing a confidence interval for a mean using the t distribution, what should you check about your data?

Accepted Answer

Check for outliers and that the population distribution is approximately normal.

Confidence Intervals and Estimation of Population Parameters

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