We wish to estimate what percent of adult residents in Ventura County like chocolate. Out of 500 adult residents sampled, 327 like chocolate. To construct a $95 \%$ confidence interval for the proportion $(p)$ of adult residents who like chocolate in Ventura County, you need to use which one of the following calculator? Hypothesis Test for a Population Proportion Two Independent Sample Means Comparison Given Data One-Way ANOVA Confidence Interval for a Population Mean Given Data Chi-Square Test for Goodness of Fit Hypothesis Test for a Population Mean Given Data Hypothesis Test for a Population Mean Given Statistics Confidence Interval for a Population Proportion Two Dependent Sample Means Comparison Given Data Two Independent Proportions Comparison Confidence Interval for a Population Mean Given Statistics Two Independent Sample Means Comparison Given Statistics Chi-Square Test for Independence a. Enter your answer as an open-interval (i.e., parentheses) using decimals (not percents) accurate to three decimal places. Confidence interval $=$ b. Express the same answer as a tri-linear inequality using decimals (not percents) accurate to three decimal places. c. Express the same answer using the point estimate and margin of error. Give your answers as decimals, to three places. \[ p=\square \pm \]
Solution
Step 1 ：We are given that out of 500 adult residents sampled, 327 like chocolate. We wish to estimate what percent of adult residents in Ventura County like chocolate.
Step 2 ：We need to construct a 95% confidence interval for the proportion (p) of adult residents who like chocolate in Ventura County.
Step 3 ：We use the formula for the confidence interval for a population proportion: \(\hat{p} \pm Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion, Z is the Z-score for the desired confidence level (for a 95% confidence level, Z is approximately 1.96), and n is the sample size.
Step 4 ：Substituting the given values into the formula, we get \(\hat{p} = 0.654\), Z = 1.96, and n = 500.
Step 5 ：We calculate the standard error (se) as \(se = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = 0.021273645667821018\).
Step 6 ：We then calculate the lower and upper bounds of the confidence interval as \(ci_{lower} = \hat{p} - Z \times se = 0.6123036544910708\) and \(ci_{upper} = \hat{p} + Z \times se = 0.6956963455089292\) respectively.
Step 7 ：Final Answer: The 95% confidence interval for the proportion of adult residents who like chocolate in Ventura County is \(\boxed{(0.612, 0.696)}\).
