The following are the P/E ratios (price of stock divided by projected earnings per share) for 15 banks:
[tex]\[ 50, 17, 15, 19, 22, 20, 23, 25, 15, 18, 43, 29, 22, 29, 22 \][/tex]

Find the [tex]30^{\text{th}}[/tex] and [tex]75^{\text{th}}[/tex] percentiles for these ratios.

(a) The [tex]30^{\text{th}}[/tex] percentile: [tex]$\square$[/tex]

(b) The [tex]75^{\text{th}}[/tex] percentile: [tex]$\square$[/tex]



Answer :

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]