In a study of the accuracy of fast food drive-through orders, Restaurant A had 242 accurate orders and 53 that were not accurate.
a. Construct a $90 \%$ confidence interval estimate of the percentage of orders that are not accurate.
b. Compare the results from part (a) to this $90 \%$ confidence interval for the percentage of orders that are not accurate at Restaurant $B: 0.156
Solution
Step 1 ：Given that Restaurant A had 242 accurate orders and 53 that were not accurate, the total number of orders is \(242 + 53 = 295\).
Step 2 ：The sample proportion \(\hat{p}\) is calculated as the number of inaccurate orders divided by the total number of orders, which is \(\frac{53}{295} = 0.180\).
Step 3 ：To construct a 90% confidence interval for the proportion of inaccurate orders, we use the formula for the confidence interval for a proportion, which is \(\hat{p} \pm Z*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion, \(Z*\) is the Z-score for the desired confidence level (for 90% confidence level, \(Z*\) is approximately 1.645), and \(n\) is the sample size.
Step 4 ：Substituting the values into the formula, we get \(0.180 \pm 1.645*\sqrt{\frac{0.180*(1-0.180)}{295}}\).
Step 5 ：Calculating the above expression, we get the 90% confidence interval for the proportion of inaccurate orders at Restaurant A as approximately \(0.143\) to \(0.216\).
Step 6 ：Given that the 90% confidence interval for the proportion of inaccurate orders at Restaurant B is \(0.156\) to \(0.236\), we see that this interval overlaps with the confidence interval for Restaurant A (\(0.143\) to \(0.216\)).
Step 7 ：Since the confidence intervals overlap, we cannot conclude that there is a significant difference between the proportions of inaccurate orders at the two restaurants.
Step 8 ：\(\boxed{\text{Final Answer:}}\)
Step 9 ：a. The 90% confidence interval for the proportion of inaccurate orders at Restaurant A is approximately \(0.143\) to \(0.216\).
Step 10 ：b. Based on the comparison of the 90% confidence intervals for the proportions of inaccurate orders at Restaurant A and Restaurant B, we cannot conclude that there is a significant difference between the two restaurants.
From Solvely APP
Solution
Step 1 ：Given that Restaurant A had 242 accurate orders and 53 that were not accurate, the total number of orders is \(242 + 53 = 295\).
Step 2 ：The sample proportion \(\hat{p}\) is calculated as the number of inaccurate orders divided by the total number of orders, which is \(\frac{53}{295} = 0.180\).
Step 3 ：To construct a 90% confidence interval for the proportion of inaccurate orders, we use the formula for the confidence interval for a proportion, which is \(\hat{p} \pm Z*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion, \(Z*\) is the Z-score for the desired confidence level (for 90% confidence level, \(Z*\) is approximately 1.645), and \(n\) is the sample size.
Step 4 ：Substituting the values into the formula, we get \(0.180 \pm 1.645*\sqrt{\frac{0.180*(1-0.180)}{295}}\).
Step 5 ：Calculating the above expression, we get the 90% confidence interval for the proportion of inaccurate orders at Restaurant A as approximately \(0.143\) to \(0.216\).
Step 6 ：Given that the 90% confidence interval for the proportion of inaccurate orders at Restaurant B is \(0.156\) to \(0.236\), we see that this interval overlaps with the confidence interval for Restaurant A (\(0.143\) to \(0.216\)).
Step 7 ：Since the confidence intervals overlap, we cannot conclude that there is a significant difference between the proportions of inaccurate orders at the two restaurants.
Step 8 ：\(\boxed{\text{Final Answer:}}\)
Step 9 ：a. The 90% confidence interval for the proportion of inaccurate orders at Restaurant A is approximately \(0.143\) to \(0.216\).
Step 10 ：b. Based on the comparison of the 90% confidence intervals for the proportions of inaccurate orders at Restaurant A and Restaurant B, we cannot conclude that there is a significant difference between the two restaurants.
