An automatic filling machine is used to fill bottles with liquid detergent. A random sample of 18 bottles results in a sample standard deviation of fill volume LaTeX: s\:=s =0.038 oz. Assume that fill volume is approximately normal. Determine the upper bound of a two-sided 97.5% confidence interval on the population variance, LaTeX: \sigma^2σ 2. Round your answer to four decimal places.

Step-by-step explanation:

Data given and notation

s=0.038 represent the sample standard deviation

represent the sample mean

n=18 the sample size

Confidence=97.5% or 0.975

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population mean or variance lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.

The Chi Square distribution is the distribution of the sum of squared standard normal deviates .

Calculating the confidence interval

The confidence interval for the population variance is given by the following formula:

The next step would be calculate the critical values. First we need to calculate the degrees of freedom given by:

Since the Confidence is 0.975 or 97.5%, the value of and , and we can use excel, a calculator or a table to find the critical values.  

The excel commands would be: "=CHISQ.INV(0.0125,17)" "=CHISQ.INV(0.9875,17)". so for this case the critical values are:

And replacing into the formula for the interval we got:

.there can be infinite values of m and n that can satisfy the above equation. for each value of m of m or n. there maybe infinite sets of solution for the above

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