Answer to Question #314370 in Statistics and Probability for Rahul

Question

7.2.2 A study by Thienprasiddhi et al. (A-4) examined a sample of 16 subjects 

with open-angle glaucoma

and unilateral hemifield defects. The ages (years) of the subjects were:

62 62 68 48 51 60 51 57

57 41 62 50 53 34 62 61

Can we conclude that the mean age of the population from which the sample 

may be presumed to have been drawn is less than 60 years? Let alpha = 0.5

Answer 1

$H_0:\mu \geqslant 60\\H_1:\mu <60\\\bar{x}=\frac{62+62+68+48+51+60+51+57+57+41+62+50+53+34+62+61}{16}=54.9375\\s^2=\frac{62^2+62^2+68^2+48^2+51^2+60^2+51^2+57^2+57^2+41^2+62^2+50^2+53^2+34^2+62^2+61^2-16\cdot 54.9375^2}{16-1}=78.7292\\s=\sqrt{s^2}=\sqrt{78.7292}=8.87295\\T=\sqrt{n}\frac{\bar{x}-\mu}{s}=\sqrt{16}\frac{54.9375-60}{8.87295}=-2.28222~t_{n-1}=t_{15}\\P-value:\\P\left( T<-2.28222 \right) =F_{T,15}\left( -2.28222 \right) =0.0187$

Since the P-value is less than the significance level, the null hypothesis is declined. The mean value is less than 60.