Mary Ann S. answered • 08/07/22
Ph.D. Educational Measurement, Doctoral Minor in Statistics.
This is a typical "central limit theorem/confidence interval/sampling distribution of the mean" problem.
1.) One complexity of this particular problem is the choice of using a Z or a single-sample t score as your test statistic. You will find much conflicting advice online. I will tell you what I typically do, and I welcome other tutors and students to comment.
a.) If the sample size is >= 30, I typically use the Z test whether or not the population standard deviation is known. (You can try a few problems out yourself to see that it doesn't make much of a difference).
b.) If the sample size is < 30, I use a t-test even if I know the population standard deviation. (I've had good results even going down to N = 25).
2.) We have a sample size of 4 in the current problem. You calculate the t-value as:
(mean(X) - hypothesized mean)/ sigma/squrt(n) =
(11.5 - 12)/.25 = -2.0
3.) The degrees of freedom for a one-sample t test are N -1, which = 3 in our case. Use excel's T.DIST() function to look up the probability of obtaining a t less than -2.0 for 3 DF. For fun, use NORM.S.DIST() to look up the probability of obtaining a Z score less than -2.0. How different are those cumulative probabilities?
For the second part of the problem, your hypothesized mean, μ = 11.25 instead of 12. I'm assuming that the sample mean remains 11.5. If the instructor wants a one-tailed test, H0 : µ = 11.25 against H1 : µ > 11.25 becomes your new alternative hypothesis. From this point, you can use the same calculations that you used in part 1.
