SOLUTION: I need help solving the following question with a calculator and how to type it in A population of values has a normal distribution with μ=103.4 and σ=63 You intend to draw a

Question 1136478: I need help solving the following question with a calculator and how to type it in
A population of values has a normal distribution with μ=103.4 and σ=63
You intend to draw a random sample of size n=242.
Find P52, which is the mean separating the bottom 52% means from the top 48% means.
P52 (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 by Theo(13342) (Show Source): You can put this solution on YOUR website!
population mean = 103
population standard deviation = 63
sample size = 242

you are looking for the probability that a particular score will have 52% of all scores in the distribution being less than it.

you need to get the standard error.

standard error = population standard deviation / square root of sample size.

se = 63 / sqrt(242) = 4.049793383

the z-score associated with an area of .52 being less than it under the normal distribution curve is equal to .0501535782.

the raw score associated with that is found by using the z-score formula of z = (x-m)/s

z is the z-score
x is the raw score
m is the raw mean
s is the standard error.

the formula becomes .0501535782 = (X-103.4)/4.049793383.

solve for x to get x = .0501535782 * 4.049793383 + 103.4 = 103.6031116.

the key is to find the standard error and use that instead of the standard deviation because you are dealing with a distribution of sample means in samples of size 242.

here's what it looks like visually with raw scores.

$$$

here's what it looks like visually with z-scores.

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