Question

determine whether the statement makes sense or does not make sense, and explain your reasoning. a poll administered to a random sample of 1150 voters shows 51% in favor of candidate a, so im 95% confident that candidate a will win the election. choose the correct answer below. a. no, this does not make sense because the margin of error of the survey is ±10.3%, which means that we can be 95% confident that between 40.7% and 61.3% of voters are in favor of candidate a. b. yes, it makes sense. c. no, this does not make sense because the margin of error of the survey is ±33%, which means that we can be 95% confident that between 18% and 84% of voters are in favor of candidate a. d. no, this does not make sense because the margin of error of the survey is ±2.9%, which means that we can be 95% confident that between 48.1% and 53.9% of voters are in favor of candidate a. e. no, this does not make sense because we can only be 51% confident that candidate a will win the election.

Explanation:

Step1: Recall margin - of - error formula

For a proportion in a large sample, the margin of error $E$ for a 95% confidence interval is given by $E = z\sqrt{\frac{p(1 - p)}{n}}$, where $z\approx1.96$ for 95% confidence, $p$ is the sample proportion, and $n$ is the sample size. Here, $n = 1150$ and $p=0.51$.

Step2: Calculate the margin of error

Substitute the values into the formula:
\[

$$\begin{align*} E&=1.96\sqrt{\frac{0.51\times(1 - 0.51)}{1150}}\\ &=1.96\sqrt{\frac{0.51\times0.49}{1150}}\\ &=1.96\sqrt{\frac{0.2499}{1150}}\\ &=1.96\sqrt{0.0002173}\\ &=1.96\times0.01474\\ &\approx 0.029 = 2.9\% \end{align*}$$

\]
The 95% confidence interval for the proportion of voters in favor of candidate A is $p\pm E=0.51\pm0.029$, or $(0.481, 0.539)$. Just because 51% of the sample is in favor does not mean we can be 95% confident candidate A will win, since the other candidate could be within the margin - of - error range.

Answer:

D. No, this does not make sense because the margin of error of the survey is $\pm2.9\%$, which means that we can be 95% confident that between 48.1% and 53.9% of voters are in favor of candidate A.