SOLUTION: I need help solving the following (not sure what steps I have to take to get the correct answer) Suppose that the distance of fly balls hit to the outfield (in baseball) is norm

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Question 1169482: I need help solving the following (not sure what steps I have to take to get the correct answer)
Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 238 feet and a standard deviation of 58 feet. We randomly sample 49 fly balls.
1. If X= average distance in feet for 49 fly balls, then what's the distribution of X? (Must round your standard deviation to two decimal places)
2. What is the probability that the 49 balls traveled an average of less than 226 feet? (Round your answer to four decimal places.)

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
mean = 238
standard deviation = 58
sample size = 49

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

standard error = 58/sqrt(49) = 58/7 = 8.285714286.

z-score = (x - m) / s

x is the raw score
m ie the mean
s is the standard error

z-score = (226 - 238) / 8.285714286 = -1.448275862

this can be rounded to -1.45 if you use the z-score tables.

it can be left as is if you use a calculator that can handle it.

i use the ti-84 plus which can handle it.

using the ti-84 plus, i get an area to the left of a z-score of -1.448275862 equal to .073769999.

that's the probability that the average of fly balls in the sample will be less than 226.

if use the z-score table, then the area to the left of a z-score of -1.45 is equal to .07353.

both results are acceptable.

the difference between .07353 and .073769999 is approximately 0.325% of .073769999.

that's well within the accuracy required.

here's what the result looks like using the online calculator that can be found at https://www.calculator.net/z-score-calculator.html?c1raw=226&c1mean=238&c1sd=8.285714286&calctype=zscore&x=60&y=17