Question 890551: A box of tickets averages out to be 75, and the SD is 10. 100 draws are made at random from this box with replacement.
A) find the chance that the average of the draws will be in the 65 to 85
B) for the range 74 to 76
I know the first one is almost 100% but im not sure about B. Is there a formula for this? The Observed Value I got is 25 and the SE is 1
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! population average is 75.
population standard deviation is 10.
sample size is 100.
standard error is equal to population standard deviation divided by square root of sample size.
that makes sample error = 10/sqrt(100) = 10/10 = 1
z score is equal to (sample mean minus population mean) divided by standard error.
to get a range, you need the high z score and the low z score and then you need to subtract the probability of getting the low z score from the probability of getting the high z score to get the probability of the range of possible values in between.
for part A, these number become:
range is 65 to 85.
z score for a sample mean of 65 would be (65 - 75) / 1 = -10.
z score for a sample mean of 85 would be (85 - 75) / 1 = +10.
the probability of getting a z score less than -10 is equal to 0.
the probability of getting a z score less than +10 is equal to 1.
the difference is 1 - 0 which is equal to 1 which is equal to 100%.
the actual probabilities are not exactly zero but they're so close to zero as to practically be zero even if they're not.
the z score tables only go to a z score of +/- 3.0 or 3.5 so they're not even close to +/- 10.
for part B, the numbers become:
z score for a sample mean of 74 is equal to (74 - 75) / 1 = -1/1 = -1.
z score for a sample mean of 76 is equal to (76 - 75) / 1 = +1/1 = +1.
the probability of getting a z score less than -1 is equal to .1587 rounded to 4 decimal places.
the probability of getting a z score less than +1 is equal to .8413 rounded to 4 decimal places.
.8413 minus .1587 = .6826 rounded to 4 decimal places = .683 rounded to 3 decimal places = .68 rounded to 2 decimal places.
you can use a z score calculator to find your answer, or you can use a z score table to find your answer.
a z score calculator that is most excellent can be found here.
http://davidmlane.com/hyperstat/z_table.html
a z score table that is quite good can be found here.
http://www.regentsprep.org/Regents/math/algtrig/ATS7/ZChart.htm
in this z score table, you look up the z score and the table shows you the probability of getting a z score less than that.
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