Question

A population of values has a normal distribution with μ = 110.6 μ = 110.6 and σ = 74 σ = 74 . You intend to draw a random sample of size n = 157 n = 157 . Find P68, which is the score separating the bottom 68% scores from the top 32% scores. P68 (for single values) = Find P68, which is the mean separating the bottom 68% means from the top 32% means. P68 (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 A population of values has a normal distribution with mean μ = 110.6 and standard deviation σ = 74, and drew a random sample of size n = 157.

a) Thus the score separating the bottom 68% scores from the top 32% scores. P68 (for single values) is calculated by finding the Z score for 68% which is the area from the left of the normal curve, the Z score is calculated using the excel formula for normal distribution which is =NORM.S.INV(0.68), thus the Z- value is obtained as 0.468.

Now using the Z score formula the single score is calculated as:

Thus the score separating the bottom 68% scores from the top 32% scores is 145.2.

b) The mean separating the bottom 68% means from the top 32% means. P68 (for sample means) is again calculated by finding the Z-score by excel formula, which was calculated above as Z = 0.468.

NOw using the Z score formula for sample means the sample mean that separates the bottom 68% is calculated as:

The mean separating the bottom 68% means from the top 32% means is 113.4

Note: Feel free to ask if query remains