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