Question

A population of values has a normal distribution with μ=161.3 and σ=31.6. You intend to draw a random sample of size n=140.

1. Find P₆, which is the score separating the bottom 6% scores from the top 94% scores.P₆ (for single values) =  
   
2. Find P₆, which is the mean separating the bottom 6% means from the top 94% means. P₆ (for sample means) =

Round to 1 decimal places. Answers obtained using exact z-scores or z-scores rounded to 2 decimal places are accepted

Please show me how to answer and solve for this problem:

Answer 1

Solution-  Given that

μ=161.3 and σ=31.6. sample of size n=140.

Step(1) -

you look up in the normal distribution table, or use a normal distribution calculator, to find the z-score that is associated with 6% of the area under the normal distribution curve that is to the left of it.

value of 6th percentile

-1.5548

Step(2) -

converting the normal random​ variables, x, to standard normal random​ variables, z, using the formula​ below, where μ is the mean and σ is the standard deviation.

z = (x-u)/σ

The z-value is -1.5548 for the 6th percentile.
z=(x-mean)/sd or (x-mean)/sigma/sqrt(n)
-1.5548 = (x-161.3) / (31.6/sqrt(140))

x = 157.12 = P₆(for single mean )

that score will have 6% of the normal distribution curve to the left of it and 94% of the normal distribution curve to the right of it.

that means the probability of getting a raw score less than 157.12 is 6% and the probability of getting a raw score greater than 157.12 is 94%.