what is the 90% confidence interval for chi squared

Question

The EPA limit on the allowable discharge of suspended solids into rivers and streams is 60 milligrams per liter (mg/l) per day. A study of water samples selected from the discharge at a phosphate mine shows that over a long period, the mean daily discharge of suspended solids is 48 mg/l, but day-to-day discharge readings are variable. State inspectors measured the discharge rates of suspended solids for n = 20 days and found s2 = 37 (mg/l)2. Find a 90% confidence interval for 𝜎2. (Round your answers to three decimal places.)

Answer 1

To calculate the 90% confidence interval for the population variance, the formula is ((n-1)s² / b) ≤ σ² ≤ ((n-1)s² / a) where a = χ² _1-α/2,n-1 and b = χ² _α/2,n-1

n = 20

mean = 48 (not needed for this question)

variance (s²) = 37

In order to determine a and b, we need to determine α

For a 90% confidence interval, 10% will not be covered in that interval, which is the α value (0.10).

a = χ² _1-α/2,n-1 = χ² _{1 - 0.10/2,n-1} = χ² _0.95,19 = 10.117

b = χ² _α/2,n-1 = χ² _0.05,19 = 30.144

You can find these values in the Chi-square (χ²) distribution table

https://sites.berry.edu/vbissonnette/wp-content/uploads/sites/21/2015/07/chisqr.pdf

Now plug these values into the equation

Upper interval = ((n-1)s² / a) = 19*37 / 10.117 = 703 / 10.117 = 69.487

Lower interval = ((n-1)s² / b) = 19*37 / 30.144 = 703 / 30.144 = 23.321

The 90% confidence interval would be (23.321, 69.487)