Question

Hydrangea is a flower that changes its color according the the pH of the soil. The flower becomes blue when the pH of the soil is too acidic (pH below 5.5), red when the pH of the soil is too basic (pH above 6.5), and purple when the pH of the soil is exactly between 5.5 to 6.5. A landowner claims that the average pH of their soil can produce perfectly consistent purple flowers with a standard deviation pH of at most 0.5. 

A flower cultivator thinks the standard deviation of the pH of the soil may be greater than 0.5, so they tested 15 random sites from the land area. Use a level of significance of α = 0.05.

Answer 1

$H_0: \sigma \le0.5$

$H_a:\sigma> 0.5$

$\chi^2=\frac{(n-1)\cdot s^2}{\sigma^2}=\frac{(15-1)\cdot 0.5^2}{0.5^2}=14$

from Chi-square Distribution Table:

for df=14:

critical value:

$\chi^2_{crit}=23.685$

Since test statistic is less than critical value, we can accept the null hypothesis.

So, the standard deviation of the pH of the soil cannot be greater than 0.5