Question

The Affordable Care Act is a healthcare reform bill that President Obama signed into law on March 23, 2010. This bill is also known as "ObamaCare". A recent survey found that only 13% of doctors support this legislation. To validate these findings, a random sample of 300 doctors was asked if they support the Affordable Care Act. A total of 54 doctors from this sample responded that they supported this bill. Which of the following statements is correct?

A.

The interval that contains 95% of the sample proportions is 0.064 and 0.196. Because the sample proportion is not between these two values, there is no support for the results of the survey.

B.

The interval that contains 95% of the sample proportions is 0.064 and 0.196. Because the sample proportion is between these two values, there is support for the results of the survey.

C.

The interval that contains 95% of the sample proportions is 0.092 and 0.168. Because the sample proportion is between these two values, there is support for the results of the survey.

D.

The interval that contains 95% of the sample proportions is 0.092 and 0.168. Because the sample proportion is not between these two values, there is no support for the results of the survey.

Answer 1

Given that, population proportion ( p ) = 0.13

sample size ( n ) = 300

sample proportion $(\hat p)$ = 54/300 = 0.18

A 95% confidence level has significance level $(\alpha)$ = 0.05 and critical value is, $Z_{\alpha/2} = 1.96$

95% confidence interval for sample proportion $(\hat p)$ is,

$p \pm Z_{\alpha/2} * \sqrt { \frac { p*(1-p)}{n}}$

$0.13 \pm 1.96 * \sqrt { \frac { 0.13*(1-0.13)}{300}}$

$0.13 \pm 0.038$

$0.13 - 0.038 < \hat p < 0.13 &plus; 0.038$

$0.092< \hat p < 0.168$

Here, sample proportion = 0.18 is not lies in above interval.

Therefore, correct answer is,

D) The interval that contains 95% of the sample proportions is 0.092 and 0.168. Because the sample proportion is not between these two values, there is no support for the results of the survey.