a. The 90% confidence interval for the percentage of orders that are not accurate at Restaurant A is 0.089 < p < 0.181.
b. This overlap suggests that we cannot conclusively say whether Restaurant A has a significantly different percentage of inaccurate orders compared to Restaurant B.
a. Constructing the 90% Confidence Interval for Restaurant A
Step 1: Calculate the sample proportion:
Let P be the sample proportion of inaccurate orders.
P = number of inaccurate orders / total number of orders
P = 52 orders / 384 orders = 0.135
Step 2: Determine the standard error:
SE = sqrt(P * (1 - P) / n)
SE = sqrt(0.135 * (1 - 0.135) / 384) ≈ 0.023
Step 3: Find the critical value (z):
We need a 90% confidence interval, so the confidence level is 1 - 0.9 = 0.1.
Since the normal distribution is symmetrical, half of the confidence level (0.05) is in each tail.
Using a z-table or calculator, we find the z-score that corresponds to an area of 0.95 in the right tail: z ≈ 1.645
Step 4: Calculate the confidence interval:
Lower bound = P - z * SE
Lower bound = 0.135 - 1.645 * 0.023 ≈ 0.089
Upper bound = P + z * SE
Upper bound = 0.135 + 1.645 * 0.023 ≈ 0.181
Therefore, the 90% confidence interval for the percentage of orders that are not accurate at Restaurant A is 0.089 < p < 0.181.
b. Comparing the Confidence Intervals for Restaurant A and B
Comparison:
Restaurant A: 0.089 < p < 0.181
Restaurant B: 0.124 < p < 0.181
The confidence intervals overlap, meaning they share some common values for the true proportion of inaccurate orders.
This overlap suggests that we cannot conclusively say whether Restaurant A has a significantly different percentage of inaccurate orders compared to Restaurant B.
More data or a different confidence level might be needed to draw a more definitive conclusion.