Find a 90% confidence interval for 𝜎2. (Round your answers to three decimal places.)

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

The upper and lower bounds of the confidence interval are defined as (n-1)s² / (chi squared value), where the chi squared values are at the 5% and 95% level. You can find these values in a table or by using a calculator.

X²_{.05, 19} = 10.117

X²_{.95, 19} = 30.144

Plugging these values into the equation we get the values 23.322 and 69.487. This is the 90% confidence interval for the variance, and it has the following meaning:

If we take 100 such samples and confidence intervals, we would expect about 90 of them to contain the true variance.

That being said, the more understandable statement would be to say that we are 90% confident that the true variance is between 23.322 and 69.487.