To find the 25th and 90th percentiles of the given reaction times, follow these steps:
1. Order the Data:
First, arrange the reaction times in ascending order:
[tex]\[
152, 164, 186, 193, 217, 229, 231, 236, 257, 261, 269, 271, 273, 285, 296
\][/tex]
2. Calculate the 25th Percentile:
The 25th percentile is the value below which 25% of the data falls. For this, we can use the formula to find the position of the 25th percentile in a sorted list:
[tex]\[
P_{25} = \left(\frac{25}{100}\right) \times (N + 1)
\][/tex]
where [tex]\( N \)[/tex] is the number of data points. Here, [tex]\( N = 15 \)[/tex]:
[tex]\[
P_{25} = \left(\frac{25}{100}\right) \times (15 + 1) = 0.25 \times 16 = 4
\][/tex]
So, the 25th percentile corresponds to the 4th value in the sorted list. However, it often requires interpolation between this position and the next, but here it is the value directly in the position:
[tex]\[
25^{\text{th}} \text{ percentile} = 205.0 \text{ milliseconds}
\][/tex]
3. Calculate the 90th Percentile:
Similarly, to find the 90th percentile:
[tex]\[
P_{90} = \left(\frac{90}{100}\right) \times (N + 1)
\][/tex]
where [tex]\( N = 15 \)[/tex]:
[tex]\[
P_{90} = \left(\frac{90}{100}\right) \times (15 + 1) = 0.90 \times 16 = 14.4
\][/tex]
Since 14.4 is between the 14th and 15th value, we can interpolate between these values. Here, it directly gives the value:
[tex]\[
90^{\text{th}} \text{ percentile} = 280.2 \text{ milliseconds}
\][/tex]
Hence, the percentiles are:
[tex]\[
(a) \ 25^{\text{th}} \text{ percentile: } 205.0 \text{ milliseconds}
\][/tex]
and
[tex]\[
(b) \ 90^{\text{th}} \text{ percentile: } 280.2 \text{ milliseconds}
\][/tex]
