Problem

On the right are the numbers of customers served by a restaurant on 40 consecutive days. (The numbers have been ranked lowest to highest.) Find the $65^{\text {th }}$ percentile.
\[
\begin{array}{llllllllll}
46 & 48 & 48 & 49 & 50 & 54 & 55 & 56 & 56 & 58 \\
59 & 61 & 61 & 62 & 63 & 66 & 66 & 68 & 69 & 70 \\
70 & 70 & 70 & 71 & 71 & 73 & 74 & 76 & 76 & 79 \\
80 & 81 & 82 & 84 & 85 & 87 & 87 & 88 & 88 & 88
\end{array}
\]

The number of customers representing the $65^{\text {th }}$ percentile is $\square$.

Answer

Expert–verified
Hide Steps
Answer

Final Answer: The number of customers representing the $65^{ ext {th }}$ percentile is \(\boxed{74}\).

Steps

Step 1 :The percentile of a set of values is the value below which a certain percent of the observations fall. For example, the 20th percentile is the value (or score) below which 20 percent of the observations may be found. The term percentile and the related term percentile rank are often used in the reporting of scores from norm-referenced tests.

Step 2 :To find the 65th percentile, we need to sort the data in ascending order, which is already done in this case.

Step 3 :Then, we find the rank of the 65th percentile using the formula: Rank = Percentile/100 * (Number of data + 1).

Step 4 :Substituting the given values into the formula, we get Rank = 65/100 * (40 + 1) = 26.65.

Step 5 :After finding the rank, if the rank is an integer, the percentile is the average of the data at position rank and rank+1. If the rank is not an integer, round it up to the nearest integer, and the percentile is the data at this position.

Step 6 :Since 26.65 is not an integer, we round it up to 27. The 27th value in the sorted data set is 74.

Step 7 :Final Answer: The number of customers representing the $65^{ ext {th }}$ percentile is \(\boxed{74}\).

link_gpt