A population of values has a normal distribution with μ = 17.1 and σ = 12.5 . You intend to draw a random sample of size n = 97 . Find P8, which is the mean separating the bottom 8% mean

Question

A population of values has a normal distribution with μ=17.1 and σ=12.5. You intend to draw a random sample of size n=97.

Find P₈, which is the mean separating the bottom 8% means from the top 92% means. 

P₈ (for sample means) = 

Answer 1

For this question, you would be looking for the value of a sample mean that corresponds to .08 area in the left tail. The first step is to determine the z value closest to .08 to the left. Using a z-table, a s value of -1.41 with a probability of .0793 is closest to .08.

We now use this x value to determine the sample mean that corresponds to the x value of -1.41

In this case we are looking at the sampling distribution. This distribution has a mean that equals the population mean for individual values. The standard deviation, however, is adjusted based on sample size. It is the standard deviation from the distribution of individual values divided by the square root of the sample size.

We use these values and rearrange the equation for the x value to solve for the x-bar value.

z=xbar - Mu/ (sigma/sq root of sample size)

solving for xbar: xbar =(sigma/sq root sample size)*z + mu

so xbar= (12.5/sqroot(97))* (-1.41) + 17.1=15.3