Question

A random sample is selected form a normal population with a mean of μ = 40...

A random sample is selected form a normal population with a mean of μ = 40 and a standard deviation of σ = 10. After a treatment is administered to the individuals in the sample, the sample mean is found to be M = 46.

a. How large a sample is necessary for this sample mean to be statistically significant? (Assume a two-tailed test with α = 0.05)

b. If the sample mean were M = 43, what sample size is needed to be significant for a two-tailed test with α = 0.05?

(a) For the sample mean of M = 46 to be statistically significant, the margin of error need to be atmost M - = 6 for 95% confidence interval

Let n be the sample size for margin of error to be 6

Critical z value corresponding to 95% CI (or two-tailed test with = 0.05) is 1.96

Thus, ≤ 6

-> ≤ 6

-> n ≥ 10.67

Thus, a sample of atleast 11 is necessary for the sample mean to be statistically significant

(b)

Similar to (a), here the margin of error needs to be atmost 43 - 40 = 3

Thus, ≤ 3

-> n ≥ 42.68

Thus, a sample of atleast 43 is necessary for the sample mean to be statistically significant