SOLUTION: Hi! I recently asked this question but could not understand the explanation of steps. I'm looking to see where the numbers in each step are coming from, could someone help me? NOTE

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Question 1138439: Hi! I recently asked this question but could not understand the explanation of steps. I'm looking to see where the numbers in each step are coming from, could someone help me? NOTE: I only need help with part B
The mean travel time to work in the US is 25.1 minutes with a standard deviation of 6.4 minutes.
a. Find the probability that a random sample of 36 people will have a mean travel time greater than 23 minutes. (0.9755)
b. Find the 90th percentile of the sample mean for those 36 people. (Answer key says 26.467, but how?!)
Thanks in advance!
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
i think i answered this for you, maybe not well enough.

the 90th percentile means that 90% of the scores are below the indicated score.

therefore, you are looking for a z-score that has 90% of hhe area under the normal distribution curve to the left of it.

i'll work with the tables rather than a calculator so you can see the process.

calculators make it easy, but can hide some of the steps behind the process.

first you look up an area of .90 in the z-score table.

the table i used for this can be found at https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf

the table shows you the area to the left of the z-score.

i looked for an area of .90.

i found the following:

$$$
$$$

what this told me is that the area of .9 to the left of the z-score is somewhere between a z-score of 1.28 and 1.29.

it is closer to 1.28 than to 1.29 and a rough interpolation would indicates a z-score somewhere around 1.282.

fortunately, i had a calculator that does that interpolation for me.

that calculator can be found at http://davidmlane.com/hyperstat/z_table.html

that calculator told me that my rough interpolation was pretty good and that the z-score associated with an area of .9 to the left of it was 1.282. as shown below.

$$$

the z-score is 1.282.

it needs to be associated with a raw score.

the z-score formula is z = (x - m) / s

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

you are given that the mean of the populaltion is 25.1 and the standard deviation of the population is 6.4.

you are also given that the sample size is 36.

the formula for standard error is s = standard deviation / square root of samaple size.

this makes s = 6.4 / sqrt(36) = 1.066667 rounded to 6 decimal digits.

the z-score formula becomes 1.282 = (x - 25.1) / 1.066667.

solve for x to get x = 1.066667 * 1.282 + 25.1 = 26.467 rounded to 3 decimal digits.

i hope this makes more sense to you.

good luck in understanding.

if you still don't understand how it's done, then send me an email with a more detailed explanation of which part of it you are still struggling with.