Problem
Use the accompanying radiation levels $\left(\right.$ in $\left.\frac{\mathrm{W}}{\mathrm{kg}}\right)$ for 50 different cell phones. Find the percentile $P_{75}$. \[ \begin{array}{llllllllll} 0.22 & 0.24 & 0.34 & 0.49 & 0.59 & 0.60 & 0.62 & 0.63 & 0.72 & 0.83 \\ 0.91 & 0.91 & 0.92 & 0.93 & 0.94 & 0.96 & 1.01 & 1.01 & 1.07 & 1.07 \\ 1.07 & 1.12 & 1.13 & 1.13 & 1.13 & 1.14 & 1.18 & 1.18 & 1.19 & 1.19 \\ 1.22 & 1.22 & 1.23 & 1.27 & 1.29 & 1.30 & 1.31 & 1.32 & 1.33 & 1.34 \\ 1.35 & 1.36 & 1.36 & 1.41 & 1.43 & 1.44 & 1.45 & 1.46 & 1.49 & 1.49 \end{array} \] $P_{75}= \pm \frac{W}{k g}$ (Type an integer or a decimal. Do not round.)
Solution
Step 1 ：First, we need to sort the data in ascending order.
Step 2 ：Then, we use the formula for the percentile, which is \(P_{k} = \frac{k(n+1)}{100}\), where \(P_{k}\) is the kth percentile and n is the number of data points. In this case, k is 75 and n is 50.
Step 3 ：Substitute the values into the formula, we get \(P_{75} = \frac{75(50+1)}{100} = 38.25\).
Step 4 ：Since the index we got is not an integer, we need to take the average of the 38th and 39th values in the sorted data set to get the 75th percentile.
Step 5 ：Finally, we find that the 75th percentile of the radiation levels is \(\boxed{1.3175} \frac{W}{kg}\).
