Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. $n = 130 , x = 69 ; 90 \%$ confidence
A) $0.463

Question

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. $n = 130 , x = 69 ; 90 \%$ confidence
A) $0.463 < p < 0.599$
B) $0.458 < p < 0.604$
C) $0.459 < p < 0.603$
D) $0.461 < p < 0.601$

Quizplus · Accepted Answer

The formula for a confidence interval for a population proportion is:
$$	ext{point estimate} \pm 	ext{margin of error}$$
where the point estimate is the sample proportion, $\hat{p} = \frac{x}{n}$, and the margin of error is:
$$	ext{margin of error} = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$
where $z^*$ is the critical value from the standard normal distribution corresponding to the given confidence level.

In this case, $n=130$, $x=69$, and the confidence level is $90\%$. The sample proportion is:
$$\hat{p} = \frac{x}{n} = \frac{69}{130} \approx 0.531$$
To find the critical value, we look at the area to the left of the critical value in the standard normal distribution table that corresponds to an area of $0.05$ (since we are using a $90\%$ confidence level, which leaves $10\%$ in the tails). Using the table, we find that $z^* \approx 1.645$.

Substituting into our formula, we get:
$$	ext{margin of error} = 1.645 \sqrt{\frac{(0.531)(1-0.531)}{130}} \approx 0.072$$
Therefore, the $90\%$ confidence interval for the population proportion is:
$$0.531 - 0.072 < p < 0.531 + 0.072$$
which simplifies to:
$$0.459 < p < 0.603$$

Thus, the answer is C, as it matches our calculated interval.

Use the Given Degree of Confidence and Sample Data to Construct