Question 1132378: 1. Is It Unusual? A population is normally distributed with a mean of 100 and a standard deviation of 15. Determine if the following event is unusual. Explain your reasoning.
a. mean of a sample of 3 is 115 or more
b. mean of a sample of 20 is 105 or more
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you need to calculate the standard error.
the standard error is the standard deviation of the distribution of sample means.
the larger the size of the sample, the smaller the standard error.
your population has a mean of 100 and a standard deviation of 15.
the standard error is equal to the standard deviation divided by the square root of the sample size.
for a sample size of 3, the standard error is equal to 15 / sqrt(3) = 8.66025...
for a sample size of 20, the standard error is equal to 15 / sqrt(20) = 3.35410...
you want to find the z-score and then determine the area to the right of that z-score.
the formula for z-wscore is z = (x - m) / s
z is the z-score
x is the raw score you are comparing to the mean.
m is the mean.
s is the standard error.
for the sample of 3, you want to find out if a score of 115 is unusual.
the formula for z-score becomes z = (115 - 100) / 8.66025 = 1.73205...
the probability of getting a z-score greater than 1.73205 ..... is .041632....
what that says is that, if you took a very large number of samples of size 3, approximately 4.2% of them would have a mean greater than 115.
conversely, approximately 95.8% of them would have a mean less than 115.
with a sample size of 20 and a score of 105 or more, the z-score would be equal to (105 - 100) / 3.35410 = 1.490712...
a probability of getting a z-score greater than 1.490712... is .068018...
what that says is that, if you took a very large number of samples of size 20, approximately 6.8% of them would have a mean greater than 105.
conversely, approximately 93.2% of them would have a mean less than 105.
are either of these unusual?
it depends on what you consider unusual.
if you consider occurring 5% of the time or less is unusual, than the first one is unusual and the second one isn't.
if you consider occurring 10% of the time or less is unusual, than both are unusual.
if you consider occurring 2% of the time or less is unusual, than neither are unusual.
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