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

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

Question 1

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

Accepted Answer

A single value calculated from sample data that serves as the best guess for a population parameter.

Question 2

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

Accepted Answer

Because it incorporates a margin of error, providing information about the accuracy of the estimate.

Question 3

Which property must a good point estimator possess?

Accepted Answer

It must be unbiased, meaning its sampling distribution is centered at the parameter it estimates.

Question 4

Suppose you want to construct a 95% confidence interval for a population proportion $$p. $$Which formula should you use for the margin of error?

Accepted Answer

$$1.96 \sqrt{\frac{p(1-p)}{n}}$$

Question 5

A survey of 1,285 people found that 31% agreed with a statement. The estimated standard deviation of the sample proportion is 0.01. What is the 95% confidence interval for the population proportion?

Accepted Answer

$$(0.29, 0.33)$$

Question 6

Which condition must be met for the normal approximation to be valid when constructing a confidence interval for a population proportion?

Accepted Answer

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

Question 7

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

Accepted Answer

The interval becomes wider.

Question 8

A clinical trial of a new flu vaccine involved 3,900 subjects, of whom 26 developed influenza. Calculate the sample proportion and the standard error for a 99% confidence interval.

Accepted Answer

Sample proportion $$= 0.0067$$, standard error $$= 0.0013$$

Question 9

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

Accepted Answer

$$\bar{x} \pm t_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)$$

Question 10

What is the degrees of freedom used in the t-distribution for constructing a confidence interval for a mean from a sample of size $$n$$?

Accepted Answer

$$df = n - 1$$

Question 11

A random sample of 11 eBay auction prices for a phone yields a sample mean of $$583.64$$ and a sample standard deviation of $$22.15. $$What is the standard error of the mean?

Accepted Answer

$$6.68$$

Question 12

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

Accepted Answer

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

Question 13

If you want a 95% confidence interval for a population mean with a margin of error $$m$$, and you estimate the population standard deviation as $$\sigma$$, what is the required sample size $$n$$?

Accepted Answer

$$n = \left( \frac{z_{\alpha/2} \sigma}{m} \$$right)^2$$

Question 14

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

Accepted Answer

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

Question 15

Which of the following factors does NOT affect the choice of sample size for a study?

Accepted Answer

The color of the data points in a graph.

Question 16

When constructing a confidence interval for a proportion with a small sample size (fewer than 15 successes or failures), what adjustment should be made?

Accepted Answer

Add 2 to the number of successes and 2 to the number of failures, increasing $$n by 4.$$

Question 17

What is the correct interpretation of a 95% confidence interval for a population parameter?

Accepted Answer

If the method is used repeatedly, about 95% of the intervals constructed will contain the true parameter.

Question 18

Which of the following is a consequence of increasing the sample size when constructing a confidence interval?

Accepted Answer

The margin of error decreases.

(Lecture 18) Statistical Inference: Point and Interval Estimation of Population Parameters

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