Question

need help with this one

Find the confidence interval for estimating the population proportion for parts (a) through (c). a. 92.5% confidence level; n = 600; P=0.20 b. 99% confidence level; n = 140; p = 0.04 C. Q=0.09; n = 375; p=0.90 Click the icon to view the standard normal table of the cumulative distribution function. a. The confidence interval is <P (Round to four decimal places as needed.) b. The confidence interval is <p<. (Round to four decimal places as needed.) C. The confidence interval is << (Round to four decimal places as needed.)

Answer 1

Solution :

a ) Given that

n = 600

$\hat p$ = 0.200

1 -$\hat p$ = 1 - 0.200 = 0.800

At 92.5% confidence level the z is ,

$\alpha$ = 1 - 92.5% = 1 - 0.925 = 0.075

$\alpha$ / 2 =0.075 / 2 = 0.0375

Z_$\alpha$/2 = Z_0.0375 = 1.780

Margin of error = E = Z_{$\alpha$ / 2} * $\sqrt$(($\hat p$ * (1 - $\hat p$)) / n)

= 1.780 * ($\sqrt$((0.200 * 0.800) / 600)

= 0.029

A 92.5 % confidence interval for population proportion p is ,

$\hat p$- E < P <$\hat p$ + E

0.200 - 0.029 < p < 0.200 + 0.029

0.171 < p < 0.229

b) Given that

n = 140

$\hat p$ = 0.040

1 -$\hat p$ = 1 - 0.040 = 0.960

At 99% confidence level the z is ,

$\alpha$  = 1 - 99% = 1 - 0.99 = 0.01

$\alpha$ / 2 = 0.01 / 2 = 0.005

Z_$\alpha$/2 = Z_0.005 = 2.576

Margin of error = E = Z_{$\alpha$ / 2} * $\sqrt$(($\hat p$ * (1 - $\hat p$)) / n)

= 2.576 * ($\sqrt$((0.040 * 0.960) /140)

= 0.043

A 99% confidence interval for population proportion p is ,

$\hat p$- E < P <$\hat p$ + E

0.040 - 0.043 < p < 0.040 + 0.043

-0.003 < p < 0.083

c ) Given that

n = 375

$\hat p$ = 0.900

1 -$\hat p$ = 1 - 0.900 = 0.100

At 91% confidence level the z is ,

$\alpha$  = 1 - 91% = 1 - 0.91 = 0.09

$\alpha$ / 2 = 0.09 / 2 = 0.045

Z_$\alpha$/2 = Z_0.045 = 2.576

Margin of error = E = Z_{$\alpha$ / 2} * $\sqrt$(($\hat p$ * (1 - $\hat p$)) / n)

= 2.576 * ($\sqrt$((0.900 * 0.100) /375)

= 0.026

A 99% confidence interval for population proportion p is ,

$\hat p$- E < P <$\hat p$ + E

0.900 - 0.026 < p < 0.900 + 0.026

0.874 < p < 0.926