Question

Use the information about each of the following samples to compute the confidence interval to estimate p.

a. n = 44 and p? = 0.50; compute a 99% confidence interval.
b. n = 300 and p? = 0.80; compute a 95% confidence interval.
c. n = 1,150 and p? = 0.40; compute a 90% confidence interval.
d. n = 95 and p? = 0.30; compute a 95% confidence interval.

Answer 1

Solution :  

Given that,

a.

n = 44

point estimate = sample proportion = = 0.50

1 - = 1 - 0.50 = 0.5

At 99% confidence level the z is ,

= 1 - 99% = 1 - 0.99 = 0.01

/ 2 = 0.01 / 2 = 0.005

Z_/2 = Z_0.005 = 2.576

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 2.576 * (((0.50 * 0.5) / 44)

= 0.194

A 99% confidence interval for population proportion p is ,

- E < p < + E

0.50 - 0.194 < p < 0.50 + 0.194

0.306 < p < 0.694

The 99% confidence interval for the population proportion p is : 0.306 , 0.694

b.

n = 300

Point estimate = sample proportion = = 0.80

1 - = 1 - 0.80 = 0.2

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z_/2 = Z_0.025 = 1.96

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.96 * (((0.80 * 0.2) / 300)

= 0.045

A 95% confidence interval for population proportion p is ,

- E < p < + E

0.80 - 0.045 < p < 0.80 + 0.045

0.755 < p < 0.845

The 95% confidence interval for the population proportion p is : 0.755 , 0.845

c.

n = 1150

point estimate = sample proportion = = 0.40

1 - = 1 - 0.40 = 0.6

At 90% confidence level the z is ,

   = 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

Z_/2 = Z_0.05 = 1.645

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.645 * (((0.40 * 0.6) / 1150)

= 0.024

A 90% confidence interval for population proportion p is ,

- E < p < + E

0.40 - 0.024 < p < 0.40 + 0.024

0.376 < p < 0.424

The 90% confidence interval for the population proportion p is : 0.376 , 0.424

d.

n = 95

Point estimate = sample proportion = = 0.30

1 - = 1 - 0.30 = 0.7

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z_/2 = Z_0.025 = 1.96

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.96 * (((0.30 * 0.7) / 95)

= 0.092

A 95% confidence interval for population proportion p is ,

- E < p < + E

0.30 - 0.092 < p < 0.30 + 0.092

0.208 < p < 0.392

The 95% confidence interval for the population proportion p is : 0.208 , 0.392