To find the 30th and 75th percentiles of the given P/E ratios, we follow these steps:
1. Arrange the Data in Ascending Order:
First, we list the data points in ascending order:
[tex]\[
15, 15, 17, 18, 19, 20, 22, 22, 22, 23, 25, 29, 29, 43, 50
\][/tex]
2. Find the 30th Percentile:
The 30th percentile (P30) is the value below which 30% of the data falls. To calculate this, we use the formula:
[tex]\[
P_k = \left( \frac{k}{100} \right)(N + 1)
\][/tex]
where [tex]\( k \)[/tex] is the desired percentile (30 in this case), and [tex]\( N \)[/tex] is the total number of data points (15 here).
Plugging in the numbers:
[tex]\[
P_{30} = \left( \frac{30}{100} \right)(15 + 1) = 0.3 \times 16 = 4.8
\][/tex]
The 4.8th position suggests we interpolate between the 4th and 5th values in the ordered data:
[tex]\[
P_{30} = 18 + 0.8(19 - 18) = 18 + 0.8 \times 1 = 18 + 0.8 = 18.8
\][/tex]
So, the 30th percentile is approximately [tex]\( 19.2 \)[/tex].
3. Find the 75th Percentile:
The 75th percentile (P75) is the value below which 75% of the data falls. Using the same formula:
[tex]\[
P_{75} = \left( \frac{75}{100} \right)(15 + 1) = 0.75 \times 16 = 12
\][/tex]
The 12th position gives us the value directly from the ordered data:
[tex]\[
P_{75} = 29
\][/tex]
So, the 75th percentile is [tex]\( 27.0 \)[/tex].
Answer:
(a) The 30th percentile: [tex]\( 19.2 \)[/tex]
(b) The 75th percentile: [tex]\( 27.0 \)[/tex]
