Question

People were polled on how many books they read the previous year. Initial survey results indicate that s= 19.3 books. Complete parts ​(a) through ​(d) below.

(a) How many subjects are needed to estimate the mean number of books read the previous year within four books with 95​% ​confidence?

This 95% confidence level requires ____subjects. ​(Round up to the nearest​ subject.)

(b) How many subjects are needed to estimate the mean number of books read the previous year within two

books with 95​% ​confidence?

This 95% confidence level requires _____subjects. ​(Round up to the nearest​ subject.)

(c) What effect does doubling the required accuracy have on the sample​ size?

A.Doubling the required accuracy nearly quadruples the sample size.

B.Doubling the required accuracy nearly halves the sample size.

C.Doubling the required accuracy nearly quarters the sample size.

D. Doubling the required accuracy nearly doubles the sample size.

​(d) How many subjects are needed to estimate the mean number of books read the previous year within four books with 99% confidence?

This 99% confidence level requires _____ subjects. ​(Round up to the nearest​ subject.)

This 99% confidence level requires ____subjects. ​(Round up to the nearest​subject.)

Compare this result to part (a). How does increasing the level of confidence in the estimate affect sample​ size? Why is this​ reasonable?

A. Increasing the level of confidence increases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a smaller sample size.

B. Increasing the level of confidence decreases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a larger sample size.

C. Increasing the level of confidence decreases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a smaller sample size.

D. Increasing the level of confidence increases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a larger sample size.

Answer 1

(a)

Sample Size (n) is given by:

2

Given:

$\alpha$ = 0.05

From Table, critical values of Z = $\pm$ 1.96

s = 19.3

e= 4

Substituting, we get:

1.962 × 19.32 n= = 90 rl

This 95% confidence level requires 90 subjects.

(b)

Sample Size (n) is given by:

2

Given:

$\alpha$ = 0.05

From Table, critical values of Z = $\pm$ 1.96

s = 19.3

e= 2

Substituting, we get:

1.962 × 19.32 358 rl

This 95% confidence level requires 358 subjects.

(c)

Correct option:

A. Doubling the required accuracy nearly quadruples the sample size.

(d)

Sample Size (n) is given by:

2

Given:

$\alpha$ = 0.01

From Table, critical values of Z = $\pm$ 2.576

s = 19.3

e= 4

Substituting, we get:

2.5762 x 19.32 155 rl

This 99% confidence level requires 155 subjects.

Correct option:

D. increasing the level of confidence increases the sample size required. For a fixed margin of error, greater confidence can be achieved with a larger sample size.