A supermarket selected a sample of 240 of its customers and

A supermarket selected a sample of 240 of its customers and measured how long they took to be served at the checkout counter. If too many customers wait too long, the supermarket intends to hire more checkout personnel. Specifically, the supermarket would like at least 86% of its customers to be checked out in 8.6 minutes or less. From the data, the 90th and 60th percentiles were computed to be 9.3 minutes and 6.7 minutes, respectively. The range of the data was 13 minutes and the fastest anyone was checked out was 1.7 minutes. What was the longest time (in minutes) anyone waited in line? Approximately how many customers waited 6.7 minutes?

Solution

The range is the difference of maximum and minimum data values. Here range is 13 minutes and the smallest value, that is minimum time taken to check out a customer is 1.7 minutes.

Such as,

Range = Max. - Min

13 = Max - 1.7

Therefore the the longest time taken in waiting is,

= 13 + 1.7 = 14.7 minutes

Since here we are considering a large data set so we can assume the given population to be normal. Let X denotes the waiting time and follows normal distributtion.

Therefore the given percentiles can be expressed as:

P ( X < 9.3 ) = 0.90

P ( X < 6.7 ) = 0.60

This implies that 90% customers waited for 9.3 minutes or less and 60% customers waited for 6.7 minutes or less .

Further, 60% of 240 is 144, that is, approximately 144 customers waited 6.7 minutes.