Question

The standard deviation of the annual precipitations of a region over a period of 100 years was 16 cm. During the last 15 years a standard deviation of annual precipitations was computed as 10 cm. Test the hypothesis that the precipitations in the region have become less variable than in the past, using a significance level of (a) 0.05 and (b) 0.01. Use chi-square distribution

Answer 1

Null hypothesis $H_0:\sigma=16.$

Alternative hypothesis $H_a:\sigma<16.$

Test statistic: $\chi^2=\frac{(n-1)s^2}{\sigma^2}=\frac{(15-1)10^2}{16^2}=5.47.$

Degrees of freedom: $df=n-1=15-1=14.$

P-value: $p=0.022.$

(a) Since the P-value is less than 0.05, reject the null hypothesis.

(b) Since the P-value is greater than 0.01, fail to reject the null hypothesis.