Question

construct the requested confidence interval.
a survey of 300 union members in new york state reveals that 112 favor the republican candidate for governor. construct a 90% confidence interval for the percentage of all new york state union members who favor the republican candidate.
(30.1%, 44.5%)
(32.7%, 41.9%)
(31.9%, 42.8%)
(26.7%, 47.9%)
(17.8%, 56.8%)

Explanation:

Step1: Calculate sample proportion

The sample proportion $\hat{p}=\frac{x}{n}$, where $x = 112$ (number of successes) and $n=300$ (sample size). So, $\hat{p}=\frac{112}{300}\approx0.373$.

Step2: Find $z -$ value

For a $90\%$ confidence interval, the significance level $\alpha=1 - 0.90=0.10$, and $\alpha/2=0.05$. The $z -$ value $z_{\alpha/2}=z_{0.05}\approx1.645$ (from standard normal distribution table).

Step3: Calculate margin of error

The margin of error $E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. Substitute $\hat{p}=0.373$, $n = 300$, and $z_{\alpha/2}=1.645$.

First, calculate $\hat{p}(1-\hat{p})=0.373\times(1 - 0.373)=0.373\times0.627 = 0.234$.

Then, $\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.234}{300}}\approx\sqrt{0.00078}\approx0.028$.

$E=1.645\times0.028\approx0.046$.

Step4: Calculate confidence interval

The confidence interval for the proportion $p$ is $\hat{p}-E

Substitute $\hat{p}=0.373$ and $E = 0.046$.

Lower limit: $0.373-0.046 = 0.327$ (or $32.7\%$)

Upper limit: $0.373 + 0.046=0.419$ (or $41.9\%$)

Answer:

(32.7%, 41.9%)