Question

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.

n = 130
x = 69; 90% confidence

a. 0.463 < p < 0.599
b. 0.458 < p < 0.604
c. 0.461 < p < 0.601
d. 0.459 < p < 0.603

Answer 1

d. 0.459 < p < 0.603

Explanation:

We have to calculate a 90% confidence interval for the proportion.

The sample proportion is p=0.531.

`p=X/n=69/130=0.531`

The standard error of the proportion is:

`\sigma_p=\sqrt{(p(1-p))/(n)}=\sqrt{(0.531*0.469)/(130)}\\\\\\ \sigma_p=√(0.001916)=0.044`

The critical z-value for a 90% confidence interval is z=1.645.

The margin of error (MOE) can be calculated as:

`MOE=z\cdot \sigma_p=1.645 \cdot 0.044=0.072`

Then, the lower and upper bounds of the confidence interval are:

`LL=p-z \cdot \sigma_p = 0.531-0.072=0.459\\\\UL=p+z \cdot \sigma_p = 0.531+0.072=0.603`

The 90% confidence interval for the population proportion is (0.459, 0.603).