Question 1185730: The average household income in a large town is $44,500, with a standard deviation of $17,200. If a SRS of 100 households is taken, find the following.
The probability that the sample mean is between $42,000 and $48,000. Round your answer to three decimal places.
he probability that the sample mean is below $43,000. Round your answer to three decimal places.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the key thing you have to do is get the standard error.
standard error = population standard deviation / square root of sample size.
in this problem:
standard deviation = 17,200
sample size = 100
s = standard error = 17200 / sqrt(100) = 1720.
m = population mean = 44,500.
z = (x - m) / s
z is the z-score
x is the sample mean
m is the population mean
s is the standard error.
if you want to find the probability that the sample mean is between 42,000 and 48,000, then you can find the z-score for 42,000 and the z-score for 48,000, and then find the probability of getting a z-score between them.
for 42,000, z = (42000 - 44500) / 1720 = -1.4535 rounded to 4 decimal places.
for 48,000, z = (48000 - 44500) / 1720 = 2.0349 rounded to 4 decimal places.
the probability of getting a z-score between -1.4535 and 2.0349 is equal to .906027 rounded to 6 decimal places.
you could also have done this directly, and gotten the same result.
the following online calculator will do the same for you, only it will round the answer to 3 decimal digits.
i used the ti-84 plus calculator to get the more detailed answer.
here's the display from the calculator to find the probability between the z-scores.
here's the display from the calculator to find the probability between the raw scores.
if you're looking for z-scores, then set the mean = 0 and standard deviation = 1.
if you're looking for raw scores, then use the mean and the standard error in the space that is marked as standard deviation.
when you are using the mean of a sample of a certain size, you have to use the standard error and not the standrad deviation.
the standard error is also a standard deviation, but it is the standard deviation of the distribution of sample means, which is different than the standard deviation of the sample itself.
here's a reference on standard error of the mean.
https://statisticsbyjim.com/hypothesis-testing/standard-error-mean/
the online statistical calculator i used can be found at https://davidmlane.com/hyperstat/z_table.html
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