In: Statistics and Probability
In a study of the accuracy of fast food drive-through orders, Restaurant A had 341accurate orders and 57 that were not accurate.
Construct a 90% confidence interval estimate of the percentage of orders that are not accurate.
a. Construct a 90% confidence interval. Express the percentages in decimal form.
------------- <p<-------------
(Round to three decimal places as needed.)
b. Compare the results from part (a) to this 90% confidence interval for the percentage of orders that are not accurate at Restaurant B: 0.131 <p< 0.182.
Given that In a study of the accuracy of fast food drive-through orders, Restaurant A had 341 accurate orders and 57 that were not accurate.
So number of total orders, n = 341 + 57
= 398
Proportion of orders that were not accurate,
= 57 / 398
= 0.143216
90% confidence interval for estimate of percentage of orders
that are not accurate =
Z
/2
*
* (1 -
) / n
Here given confidence level = 90% = 0.9
= 1 - confidence level
= 1 - 0.9
= 0.1
/2
= 0.05
So Z/2
will be z-score that has an area of 0.05 to its right which is
1.64485
90% confidence interval = 0.143216
1.64485 *
0.143216 * (1 - 0.143216) / 398
= 0.143216
1.64485 *
0.000308
= 0.143216
1.64485 * 0.017559
= 0.143216
0.028881
= (0.114335, 0.172097)
So 90% confidence interval for estimate of the percentage of orders that are not accurate = (0.114335, 0.172097)
After roudning to 3 decimals we have
0.114 < p < 0.172
Question (b)
As you can observe the lower limit in the confidence interval of the Restaurant A is larger than that of lower limit in the confidence interval of the Restaurant B
The higher limit in the confidence interval of the Restaurant B is smaller than that of higher limit in the confidence interval of the Restaurant B
Margin of error = (Width of the confidence interval) / 2 = (upper limit - lower limit) / 2
Margin of error for restaurant A = (0.172 - 0.114) / 2
= 0.058 / 2
= 0.029
Margin of error for restaurant B = (0.182 - 0.131) / 2
= 0.051 / 2
= 0.0255
The Margin of error for Restaurant A is more than that of Restuarant B which implies the confidence in restaurant A is less than that of the confidence in restaurant B
In other terms the accuracy of the population parameters will be more in restaurant B than in restaurant A