To estimate the mean score ? of those who took the Medical College Admission Test on your campus, you will obtain the scores of a random sample of students. From published information, you know that the scores are approximately Normal with standard deviation about 6.5. You want your sample mean (x ?) to estimate ? with an error of no more than 1 point in either direction. 1)What standard deviation must x ? have so that 99.7% of all samples give an x ? within 1 point of ?. 2) How large a random sample do you need in order to reduce the standard deviation of x ? to the value you found in part 1?
Here as 99..7% of all samples must have sample mean withing 1 point of sample mean.
so margin of error = 1
Population standard deviation = 6.5
margin of error = critical test statistic * standard error of sample mean
critical test statistic = 3(as 99.7% of samples will lie in between these values)
standard deviation of = 1/3 = 0.3333
(2) Here if the required sample size n = n
standard error of sample mean = 6.5/sqrt(n)
6.5/sqrt(n) = 0.3333
sqrt(n) = 6.5 * 3
n = 380.25
n = 381
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