If $\mathrm{n}=320$ and $\hat{p}$ ( $p$-hat) $=0.7$, construct a $95 \%$ Confidence Interval. In order to use technology, you need to compute the number of successes: $\mathrm{x}=\mathrm{n}{ }^{*} \hat{p}$ Give your answers to three decimals. \[ <\mathrm{p}< \]
Solution
Step 1 ：Given that the sample size, \(n = 320\), and the sample proportion, \(\hat{p} = 0.7\). We are asked to construct a 95% confidence interval for the population proportion.
Step 2 ：First, we need to calculate the number of successes, which is given by \(x = n \times \hat{p}\). Substituting the given values, we get \(x = 320 \times 0.7 = 224\).
Step 3 ：Next, we calculate the standard error, which is given by the formula \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\). Substituting the given values, we get \(\sqrt{\frac{0.7 \times (1 - 0.7)}{320}} = 0.0256\).
Step 4 ：The z-score for a 95% confidence interval is 1.96. We use this to calculate the lower and upper bounds of the confidence interval. The lower bound is given by \(\hat{p} - z \times se\) and the upper bound is given by \(\hat{p} + z \times se\). Substituting the given values, we get the lower bound as \(0.7 - 1.96 \times 0.0256 = 0.650\) and the upper bound as \(0.7 + 1.96 \times 0.0256 = 0.750\).
Step 5 ：Final Answer: The 95% confidence interval for the population proportion is \(\boxed{(0.650, 0.750)}\).
