Question

construct a 90% confidence interval of the population proportion using the giver information x=74 n=150

Answer 1

Answer:

The 90% confidence interval of the population proportion is (0.43, 0.56).

Step-by-step explanation:

The (1 - α)% confidence interval for population proportion p is:

$CI=\hat p\pm z_{\alpha/2}\ \sqrt{\frac{\hat p(1-\hat p)}{n}}$

The information provided is:

X = 74

n = 150

Confidence level = 90%

Compute the value of sample proportion as follows:

$\hat p=\frac{X}{n}=\frac{74}{150}=0.493$

Compute the critical value of z for 90% confidence level as follows:

$z_{\alpha/2}=z_{0.10/2}=z_{0.05}=1.645$

*Use a z-table.

Compute the 90% confidence interval of the population proportion as follows:

$CI=\hat p\pm z_{\alpha/2}\ \sqrt{\frac{\hat p(1-\hat p)}{n}}$

     $=0.493\pm 1.645\times \sqrt{\frac{0.493(1-0.493)}{150}}\\\\=0.493\pm 0.0672\\\\=(0.4258,\ 0.5602)\\\\\approx (0.43,\ 0.56)$

Thus, the 90% confidence interval of the population proportion is (0.43, 0.56).