In a large university the average age of all the students is 24 years with a standard deviation of 9 years. a random sample of 36 students is selected.
a.determine the standard error of the mean.
b.what is the probability that the sample mean will be larger than 19.5?
c.what is the probability that the sample mean will be between 25.5 and 27 years?

a. The standard error of the mean is the standard deviation divided by the square root of the sample size. Here, that's 9/√36 = 9/6 = 1.5 (years).

b. 19.5 represents 3 standard deviations below the mean. The empirical rule tells you that 99.7% of sample means will be within 3 standard deviations of the mean, so the remaining 0.3% are above or below those values. Here, that means 0.15% of sample means are below 19.5 years, so 99.85% will be above 19.5 years. A statistics calculator gives the slightly refined value 99.87%.

c. The given limits represent 1 and 2 standard deviations above the mean. Again, the empirical rule tells you about 95% -68% = 27% will fall within that band above or below the mean. We're concerned with the half of that quantity that is above the mean. About 13.5% of sample means will fall between 25.5 and 27. Again, a statistics calculator gives a slightly refined value for that: 13.6%.

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See the second attachment for the calculator's results.