Question

the affordable care act (aca) signed into law by president barrack obama in 2010 was the most significant regulatory reform of the united states healthcare system since the passage of medicare and medicaid in 1965. the law was designed to expand insurance coverage and reduce the costs of health care. some elements of the law such are banning insurance companies from denying coverage to people with pre - existing conditions are popular while other provisions such as the individual mandate are unpopular. opinions on the law vary with some wanting to keep the law as it it, others favor fixing the law and some favor repealing the law in its entirety. the middle position has been gaining traction recently and in a rasmussen reports poll conducted january 5, 2017, 527 out of 941 randomly selected likely voters want congress and the president to fix the law a piece at a time.

a. find the point estimate for the proportion of americans who want congress and the president to fix the law a piece at a time. round your answer to 4 decimal places.

b. construct the 95% confidence interval for the proportion of americans who want congress and the president to fix the law a piece at a time. enter your answer as an open interval of the form (a,b) and round to 4 decimal places.

c. find the margin of error for a 95% confidence interval. (round your answer to 4 decimal places.

d. can you conclude that more than half of americans want congress and the president to fix the law a piece at a time?
no conclusions can be drawn since the confidence interval contains 0.50.
yes, the entire confidence is above 0.50.
no, the entire confidence interval is below 0.50.

Explanation:

Step1: Calculate point - estimate

The point - estimate $\hat{p}$ for a proportion is given by $\hat{p}=\frac{x}{n}$, where $x$ is the number of successes and $n$ is the sample size. Here, $x = 527$ and $n=941$. So, $\hat{p}=\frac{527}{941}\approx0.5600$.

Step2: Calculate $q$

We know that $q = 1-\hat{p}$. So, $q=1 - 0.5600=0.4400$.

Step3: Find $z$ - value for 95% confidence interval

For a 95% confidence interval, the $z$ - value $z_{\alpha/2}=1.96$.

Step4: Calculate margin of error

The margin of error $E$ for a proportion is $E = z_{\alpha/2}\sqrt{\frac{\hat{p}q}{n}}$. Substituting the values: $\hat{p}=0.5600$, $q = 0.4400$, $n = 941$, and $z_{\alpha/2}=1.96$.

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Step5: Calculate confidence interval

The 95% confidence interval for the proportion $p$ is given by $(\hat{p}-E,\hat{p} + E)$. So, $(0.5600 - 0.0317,0.5600+0.0317)=(0.5283,0.5917)$.

Answer:

a. $0.5600$
b. $(0.5283,0.5917)$
c. $0.0317$
d. Yes, the entire confidence is above 0.50.