A supermarket selected a sample of 230 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 81% of its customers to be checked out in 8.1 minutes or less. From the data, the 90th and 60th percentiles were computed to be 9.7 minutes and 7.2 minutes, respectively. The range of the data was 13 minutes and the fastest anyone was checked out was 1.4 minutes.
1. What was the longest time (in minutes) anyone waited in line?
2. Approximately how many customers waited 7.2 minutes?

The key to answering this question is to manage the concepts of range and percentile in a distribution of data. We can roughly say that range is the difference between the highest and the lowest values in the distribution (a measure of dispersion), and percentile tells us what percentage of the data is below it.

The question says nothing about the kind of distribution the data follow. However, with the given information, we can overcome the question.

1. What was the longest time (in minutes) anyone waited in line?

According to the description in the question "the range of the data was 13 minutes and the fastest anyone was checked out was 1.4 minutes".

This gives us a range of values from 1.4 minutes to 13 minutes for time customers wait for being checked, in which 1.4 minutes was the fastest in being checked and 13 minutes the longest time in being checked.

Therefore, the longest time waited in line was 13 minutes.

2. Approximately how many customers waited 7.2 minutes?

A 60th percentile indicates that 60% of the observations are below it, a 50th percentile indicates that 50% of the observations are lower than it, and so on. That is, a percentile is a way to say the percentage of observations that are below it.

From the question, we already know that 90th and 60th percentiles are 9.7 minutes and 7.2 minutes, respectively. In other words, the 90th percentile is 9.7 minutes and the 60th percentile is 7.2 minutes. For the latter, we can say that 60% of the customers were attended below 7.2 minutes. Then, knowing that the sample was of 230 customers, we can determine the number of customers that waited for less than 7.2 minutes (60th percentile) as follows:

[tex] \\ Number\;of\;customers = 230*60\% = 230*\frac{60}{100} = 230*0.60[/tex]

[tex] \\ Number\;of\;customers = 138[/tex]

Thus, according to the definition of percentile, the number of customers that waited for less than 7.2 minutes is 138. However, we cannot say anything for those that wait exactly 7.2 minutes, only those that waited lower than 7.2 minutes (for example, 7.19 minutes). In this way, we can say that the number of customer that waited 7.2 minutes is, approximately, 138.