Question

In a study of the accuracy of fast food​ drive-through orders, Restaurant A had 206 accurate orders and 57 that were not accurate. a. Construct a 95​% confidence interval estimate of the percentage of orders that are not accurate. b. Compare the results from part​ (a) to this 95​% confidence interval for the percentage of orders that are not accurate at Restaurant​ B: 0.198less thanpless than0.299. What do you​ conclude? a. Construct a 95​% confidence interval. Express the percentages in decimal form. nothingless thanpless than nothing ​(Round to three decimal places as​ needed.) b. Choose the correct answer below. A. Since the upper confidence limit of the interval for Restaurant B is higher than both the lower and upper confidence limits of the interval for Restaurant​ A, this indicates that Restaurant B has a significantly higher percentage of orders that are not accurate. B. No conclusion can be made because not enough information is given about the confidence interval for Restaurant B. C. Since the two confidence intervals​ overlap, neither restaurant appears to have a significantly different percentage of orders that are not accurate. D. The lower confidence limit of the interval for Restaurant B is higher than the lower confidence limit of the interval for Restaurant A and the upper confidence limit of the interval for Restaurant B is also higher than the upper confidence limit of the interval for Restaurant A.​ Therefore, Restaurant B has a significantly higher percentage of orders that are not accurate.

Answer 1

Solution:

Given: Restaurant A had 206 accurate orders and 57 that were not accurate.

n = 206+ 57 = 263

Part a) Construct a 95​% confidence interval estimate of the percentage of orders that are not accurate.

Formula:

where

and

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

Thus

Part b) Compare the results from part​(a) to this 95​% confidence interval for the percentage of orders that are not accurate at Restaurant​ B: 0.198 < p < 0.299. What do you​ conclude?

Since lower limit of 95​% confidence interval for the percentage of orders that are not accurate at Restaurant​ B is within the limits of 95​% confidence interval for the percentage of orders that are not accurate at Restaurant A, thus we can say that both the confidence intervals for Restaurant A and Restaurant B overlap.

Thus we conclude that:
C. Since the two confidence intervals overlap, neither restaurant appears to have a significantly different percentage of orders that are not accurate.