SOLUTION: A normal population has a mean of 75 and a standard deviation of 5. You select a sample of 40. Compute the probability the sample mean is: a. Less than 74. b. Between 74 and 76.

Question 648028: A normal population has a mean of 75 and a standard deviation of 5. You select a sample of 40. Compute the probability the sample mean is:
a. Less than 74.
b. Between 74 and 76.
c. Between 76 and 77.
d. Greater than 77.

I would love to get the answer so I can print it out and use it as a guide.
Thank you so much!
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
population mean = 75
population standard deviation = 5
sample size = 40
i'll round all intermediate results to 3 decimal places and found the final answer to 2 decimal places.
z-score = actual score minus population score divided by standard error.
standard error = population standard deviation divided by square root of sample size.
for this problem, standard error = 5/sqrt(40) = .791
first you need to find the z-score and then you need to find the area under the normal distribution curve that applies.
use of a z-score table is mandatory.
there are different types, but the type show in this link is usually the best type to use because it's fairly easy to figure out what you need.
http://lilt.ilstu.edu/dasacke/eco148/ztable.htm
you need to compute the probability that the score is:
-----
a. Less than 74.
z-score = (74-75)/.791 = -1/.791 = -1.264 which can be rounded to -1.26 since the accuracy of the z-score tables is only 2 decimal digits.
you look into the z-score to find the area under the distribution curve that is to the left of a z-score of -1.26.
you look down the first column until you find -1.2 and then you look for the entry in the 7th column on that row.
the entry you will find is .1038.
that the area under the normal distribution curve to the left of a z-score of -1.26.
this means that the probability of getting a z-score less than or equal to -1.26 is equal to .1038 or 10.38%.
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b. Between 74 and 76.
you need to find 2 z-scores.
you need to find a z-score for an actual score of 74 and a z-score for an actual score of 76.
z-score for 74 is:
(74-75)/.791 = -1/.791 = -1.26 (rounded to 2 decimal digits).
z-score for 76 is:
(76-75)/.791 = 1/.791 = 1.26 (rounded to 2 decimal digits).
you want the area under the normal distribution curve that is between a z-score of -1.26 and a z-score of 1.26
that would be the area under the normal distribution curve to the left of a z-score of 1.26 minus the area under the normal distribution curve to the left of a z-score of -1.26.
area to the left of a z-score of 1.26 is equal to .8962
area to the left of a z-score of -1.26 is equal to .1038
.8962 minus .1038 equals .7924
that's the area between a z-score of -1.26 and 1.26 which is equal to the probability that you will get a z-score between -1.26 and +1.26 which is equal to the probability that you will get an actual score between 74 and 76 if the population mean is 75 and the population standard deviation is 5 and your sample size is 40.
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c. Between 76 and 77.
this is done in the same manner as between 74 and 76.
you get the z-score for 76 and a z-score of 77.
you then look up the area in the z-score table for a z-score of 76 and a z-score of 77.
you then subtract the area to the left of a z-score of 76 from the area to the left of a z-score of 77 to get the difference which is the area between them which is the probability of getting a z-score between them.
the numbers come out as follows:
z-score of 76 = (76-75)/.791 = 1/.791 = 1.26
z-score of 77 = (77-75)/.791 = 2/.791 = 2.53
area to the left of a z-score of 1.26 = .8962
area to the left of a z-score of 2.53 = .9943
area between a z-score of 76 and 77 = .9943 - .8962 = .0981
probability of getting a z-score between 1.26 and 2.53 = .0981
that's the probability of getting an actual score between 76 and 77 if the population mean is 75 and the population standard deviation is 5 and the sample size is 40.
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d. Greater than 77.
here you want to find the area to the left of a z-score for 77 and then take 1 minus that to get the area to the right of that z-score.
the z-score is (77-75)/791 = 2.53
the area to the left of that z-score is .9943
1 - .9943 = .0057
that's the probability of getting a z-score greater than 2.53 which is the probability of getting a raw score greater than 77 if the population mean is 75 and the population standard deviation is 5 and the sample size is 40.
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you are dealing with a sample mean and a distribution of sample means.
the raw scores above are the sample means.
the popuation mean is 75
the population standard deviation is 5
the sample size is 40.
the standard error is equal to 5/sqrt(40) which is equal to .791
the standard error is the standard deviation of the distribution of sample means.
it is affected by the sample size.
the larger the sample size, the smaller the standard error.
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remember - difference s-score tables work different ways.
the one i showed you is one of the easiest to follow.
always check the table you are using to find out what they are assuming because they don't all work the same way.

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