Question

A researcher is interested in knowing whether early childhood education has an impact on graduate earnings. She finds 14 published studies that have tested the null hypothesis that there is no impact. Since the studies were conducted at different times in different countries, it is reasonable to assume that the samples are statistically independent.

Assuming that there is in fact no relationship between early childhood education and graduate earnings, what is the probability that at least one of the 14 studies will reject the null hypothesis at a 5% significance level?

Answer 1

Significance level = 5%

P[ She will reject the null hypothesis | no relationship is there ( it is true ) ] = 5% = 0.05

P[ She will not reject the null hypothesis | no relationship is there ( it is true ) ] = 1 She will reject the null hypothesis | no relationship is there ( it is true ) ] = 1 - 0.05 = 0.95

Number of studies = n = 14

P[ At least one of them will be rejected ] = 1 - P[ None is rejected ]

P[ None is rejected ] = P[ She will not reject the null hypothesis | no relationship is there ( it is true ) ]^n

P[ None is rejected ] = 0.95^14 = 0.4877

P[ At least one of them will be rejected ] = 1 - P[ None is rejected ]

P[ At least one of them will be rejected ] = 1 - 0.4877

P[ At least one of them will be rejected ] = 0.5123