Percentiles
On each trial of an experiment, a subject is presented with a constant soft noise, which is interrupted at some unpredictable time by a noticeably louder sound. The time it takes for the subject to react to this louder sound is recorded. The following list contains the reaction times (in milliseconds) for the 19 trials of this experiment:
\[
202,186,171,250,175,261,241,229,235,212,193,226,296,285,311,236,152,164,222
\]
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Find $25^{\text {th }}$ and $70^{\text {th }}$ percentiles for these reaction times.
(If necessary, consult a list of formulas.)
(a) The $25^{\text {th }}$ percentile: $\square$ milliseconds
(b) The $70^{\text {th }}$ percentile: $\square$ milliseconds

Step 1 ：Given the reaction times in milliseconds: 202, 186, 171, 250, 175, 261, 241, 229, 235, 212, 193, 226, 296, 285, 311, 236, 152, 164, 222.

Step 2 ：First, sort the data in ascending order: 152, 164, 171, 175, 186, 193, 202, 212, 222, 226, 229, 235, 236, 241, 250, 261, 285, 296, 311.

Step 3 ：The total number of data points, \(N\), is 19.

Step 4 ：To find the 25th percentile, calculate \(P_{25} = \frac{25}{100} \times (N + 1) = 5\).

Step 5 ：To find the 70th percentile, calculate \(P_{70} = \frac{70}{100} \times (N + 1) = 14\).

Step 6 ：Since the results are not integers, interpolate between the two closest data points for each percentile.

Step 7 ：The 25th percentile, \(P_{25}\), is 186 milliseconds.

Step 8 ：The 70th percentile, \(P_{70}\), is 241 milliseconds.

Step 9 ：Final Answer: (a) The 25th percentile: \(\boxed{186}\) milliseconds (b) The 70th percentile: \(\boxed{241}\) milliseconds