Question

A population of values has a normal distribution with μ=65.3μ=65.3 and σ=42.4σ=42.4. You intend to draw a random sample of size n=242n=242.

Find P₃₄, which is the mean separating the bottom 34% means from the top 66% means.
P₃₄ (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

Given, the population is normally distributed, X ~ N( ) = N(65.3, 42.4²)

Now, when we draw a sample of size n = 242 from this population, the sample is also normally distributed with,

Mean,   = 65.3, and std deviation, s = / sqrt(n) = 42.4 / sqt(242) = 2.726

Hence, the sample has a normal distribution, Y ~ N(65.3, 2.726²)

Hence, P₃₄ = 34th percentile of the sample distribution = y (say)

=> P(Y <= y) = 0.34

=> P(Z <= (y-65.3) / 2.726 ) = 0.34 (converting to std Z-score probability)

=> (y - 65.3) / 2.726 = -0.41

=> y = 64.18

Hence, P₃₄ = 64.18