Question

A pizza shop claims its average home delivery time is 27 minutes. A sample of 30 deliveries had a sample average of 30.7 minutes. Assume the population standard deviation for the​ shop's deliveries is 7.1 minutes. Complete parts a and b below. a. Is there support for the​ shop's claim using the criteria that the sample average of 30.7 minutes falls within the symmetrical interval that includes​ 95% of the sample means if the true population mean is 27 ​minutes? Select the correct choice below and fill in the answer box within your choice. ​(Type an integer or decimal rounded to four decimal places as​ needed.) A. The probability Upper P left parenthesis x overbar greater than or equals 30.7 right parenthesisequals nothing. This indicates that the sample average of 30.7 minutes falls within the​ 95% symmetrical​ interval, supporting the​ shop's claim. B. The probability Upper P left parenthesis x overbar greater than or equals 30.7 right parenthesisequals 0.0022. This indicates that the sample average of 30.7 minutes falls outside the​ 95% symmetrical​ interval, contradicting the​ shop's claim. Your answer is not correct. b. Identify the symmetrical interval that includes​ 95% of the sample means if the true population mean is 27 minutes.

Answer 1

a) P( > 30.7)

= P(( - )/() > (30.7 - )/())

= P(Z > (30.7 - 27)/(7.1/))

= P(Z > 2.85)

= 1 - P(Z < 2.85)

= 1 - 0.9978

= 0.0022

Option - B) P( > 30.7) = 0.0022 . This indicates that the sample average of 30.7 minutes falls outside the 95% symmetrical interval, contradicting the shop's claim.

b) P( < x) = 0.025

Or, P(( - )/() < (x - )/()) = 0.025

Or, P(Z < (x - 27)/(7.1/)) = 0.025

Or, (x - 27)/(7.1/) = -1.96

Or, x = -1.96 * 7.1/ + 27

Or, x = 24.459

P( > x) = 0.025

Or, P(( - )/() > (x - )/()) = 0.025

Or, P(Z > (x - 27)/(7.1/)) = 0.025

Or, P(Z < (x - 27)/(7.1/)) = 0.975

Or, (x - 27)/(7.1/) = 1.96

Or, x = 1.96 * 7.1/ + 27

Or, x = 29.541

So, the interval is from 24.459 to 29.541.