Integrate the appropriate chisquare density to find the prob

Integrate the appropriate chi-square density to find the probability that the variance of a random sample of size 5 from a normal population with sigma^2 = 25 will fall between 20 and 30.

The answer is 0.216

You know that the sample variance s2 of a sample of N independent normals is distributed as

(N?1)s2?2??2N?1

So scale your two bounds appropriately and use the CDF of the ?2N?1 distribution.

20?42025=3.2

30?43025=4.8