Question

A random sample of 100 observations results in 70 successes. [You may find it useful to reference the z table.]

a. Construct the a 95% confidence interval for the population proportion of successes. (Round intermediate calculations to at least 4 decimal places. Round "z" value and final answers to 3 decimal places.)  

Confidence interval to

b. Construct the a 95% confidence interval for the population proportion of failures. (Round intermediate calculations to at least 4 decimal places. Round "z" value and final answers to 3 decimal places.)

Confidence interval to

Answer 1

Solution :

(a)

Given that,

n = 100

x = 70

= x / n = 70 / 100 = 0.70

1 - = 1 - 0.70 = 0.30

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 * (((0.70 * 0.30) / 100)

= 0.09

A 95% confidence interval for population proportion p is ,

- E < P < + E

0.700 - 0.610 < p < 0.700 + 0.610

0.610 < p < 0.790

Confidence interval is:  0.610 to 0.790 .

(b)

n = 100

x = 30

= x / n = 30 / 100 = 0.30

1 - = 1 - 0.30 = 0.70

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 * 0.70) / 100)

= 0.09

A 95% confidence interval for population proportion p is ,

- E < P < + E

0.300 - 0.09 < p < 0.300 + 0.09

0.210 < p < 0.390

Confidence interval is:  0.210 to 0.390