Question

A population of values has a normal distribution with $\mu =144.2$ and $\sigma =96.9$. You intend to draw a random sample of size $n=47$.

Find P₃₅, which is the mean separating the bottom 35% means from the top 65% means.
P₃₅ (for sample means) =

Enter your answers as numbers accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

Solution:

We are given that a random variable follows a normal distribution.

We are given

µ = 144.2

σ = 96.9

n = 47

We have to find 35^th percentile or P35 which is given as below:

Z-critical value for bottom 35% area or top 65% area is given as below:

Z = -0.38532 (by using normal Z-table or excel)

Formula for finding percentile is given as below:

Percentile = X = µ + Z* σ

Percentile = 144.2 + (-0.38532)*96.9

Percentile = 144.2 - 0.38532*96.9

Percentile = 106.8625

Required Percentile = P₃₅ = 106.9 (rounded to one decimal place)