Question 1204071: what is the sollution of a random sample of 27 observations from a large population has a mean of 22 and a
0.01 standard deviation of 4.8. Can we conclude at that the population mean is
significance below 24?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! sample mean is 22.
sample standard deviation is 4.8
test mean is 24.
you are looking to see if the sample mean of 22 is significantly below the test mean of 24.
the sample is taken from the poopulation, so the sample mean is a proxy for the population mean.
if the sample mean is significantly below 24, you can conclude that there is a strong likelihood that the population mean is also significantly below 24.
t-score is used because the sample size is less than 30 and the sample standard deviation is used rather than the population standard deviation, which is presumably not known.
stsndard error is used rather than standard deviation because you are comparing the mean of a sample of more than one element, rather than the score of a sample of one element.
formula to use is t = (x-m)/s
t is the t-score
x is the sample mean of 22
m is the test mean of 24
s is the standard error, which is equal to standard deviation divided by square root of sample size = 4.8 / sqrt(27) = .923760.
solve for t to get:
t = (22 - 24) / .923760 = -2.165064.
degrees of freedom = 27 - 1 = 26.
area to the left of that z-score with 26 degrees of freedom = .01987167.
that's your test alpha.
since you are looking to see if the population mean is less than the test mean, all of the critical alpha is on the left side of the confidence interval.
if your critical alpha is .05, then the test alpha indicates that the results are significant because it is less than that.
in that case, you would conclude that the population mean is less than the test mean of 24.
if your critical alpha is .01, then the test alpyha indicates that the resulw are not significant because it is greater than that.
in that case, you would conclude that there is not sufficient evidence to conclude that the population mean is less than the test mean of 24.
here's what the results look like on an online t-test calculator.
first results are at critical alpha of .01.
second results are at critical alpha of .05.
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