Answer to Question #314432 in Statistics and Probability for Rahul

Question

In a sample of 49 adolescents who served as the subjects in an

immunologic study, one variable of

interest was the diameter of skin test reaction to an antigen. The sample mean

and standard deviation

were 21 and 11mm erythema, respectively. Can it be concluded from these

data that the population

mean is less than 30? Let alpha = 0.5

Answer 1

$H_0:\mu \geqslant 30\\H_1:\mu <30\\T=\sqrt{n}\frac{\bar{x}-\mu}{s}=\sqrt{49}\frac{21-30}{11}=-5.72727\sim t_{n-1}=t_{48}\\P-value:\\P\left( T<-5.72727 \right) =F_{t,48}\left( -5.72727 \right) =3.26\cdot 10^{-7}$

Since the P-value is less than the significance level, the null hypothesis is rejected. The mean value is less than 30.