Question

Questions 19 and 20 related to the following: The lower and upper end of a 95% interval estimate of the population proportion are, respectively, 0.556 and 0.604.

The point estimate to build this interval is,
0.57
0.575
0.58
0.592
The sample size to build this interval estimate is,
1625
1545
1456
1358

Answer 1

Solution :

95% confidence interval = (0.556 and 0.604.)

1) Point estimate = $\hat p$ = (Lower confidence interval + Upper confidence interval ) / 2

Point estimate = $\hat p$ = (0.556 + 0.604) / 2

Point estimate = $\hat p$ = 0.58

Margin of error = E = Upper confidence interval - $\hat p$

Margin of error = E = 0.604 - 0.58 = 0.024

2) $\hat p$ = 0.58

1 - $\hat p$ = 1 - 0.58 = 0.42

margin of error = E = 0.024

At 95% confidence level

$\alpha$ = 1 - 95%

$\alpha$ =1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = 1.96

sample size = n = (Z_{$\alpha$ / 2} / E )² * $\hat p$ * (1 - $\hat p$ )

= (1.96 / 0.024)² * 0.58 * 0.42

= 1624.67

sample size = n = 1625