Question

A Gallup poll asked 1200 randomly chosen adults what they think the ideal number of children for a family is. Of this sample, 53% stated that they thought 2 children is the ideal number.

Answer 1

A Gallup poll asked 1200 randomly chosen adults what they think the ideal number of children for a family is. Of this sample, 53% stated that they thought 2 children is the ideal number. Construct and interpret a 95% confidence interval for the proportion of all US adults that think 2 children is the ideal number.

Answer:

at 95% Confidence interval level: 0.501776   < p < 0.558224

Step-by-step explanation:

sample size n = 1200

population proportion $\hat p$= 53% - 0.53

At 95% confidence interval level;

level of significance ∝ = 1 - 0.95

level of significance ∝ = 0.05

The critical value for $z_{\alpha/2} = z _{0.05/2}$

the critical value $z _{0.025}= 1.96$ from the standard normal z tables

The standard error S.E for the population proportion can be computed as follows:

$S,E = \sqrt{\dfrac{\hat p \times (1-\hat p)}{n}}$

$S,E = \sqrt{\dfrac{0.53 \times (1-0.53)}{1200}}$

$S,E = \sqrt{\dfrac{0.53 \times (0.47)}{1200}}$

$S,E = \sqrt{\dfrac{0.2491}{1200}}$

$S,E = 0.0144$

Margin of Error E= $z_{\alpha/2} \times S.E$

Margin of Error E= 1.96 × 0.0144

Margin of Error E= 0.028224

Given that the confidence interval for the proportion  is  95%

The lower and the upper limit for this study is as follows:

Lower limit: $\hat p - E$

Lower limit: 0.53 - 0.028224

Lower limit: 0.501776

Upper limit: $\hat p + E$

Upper limit: 0.53 + 0.028224

Upper limit: 0.558224

Therefore at 95% Confidence interval level: 0.501776   < p < 0.558224