Question

A population of values has a normal distribution with μ=230.5μ=230.5 and σ=65.4σ=65.4. You intend to draw a random sample of size n=40n=40.

Find P₂₇, which is the mean separating the bottom 27% means from the top 73% means.
P₂₇ (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.

Please provide step by step if possible!

Answer 1

Given that,

mean = = 230.5

standard deviation = = 65.4

n = 40

=   = 230.5

= / n = 65.4/ 40 = 10.34

Using standard normal table

P(Z < z) = 27%

P(Z < z ) = 0.27

P(Z < -0.613 ) = 0.27

z = -0.613

Using z-score formula

= z * +

= -0.613 * 10.34 + 230.5

= 224.2

P27 = 224.2