Distribution

We may often need to estimate confidence interval for the population variance (or standard deviation).

Like every sample statistic, the sample variance is a random variable and it possesses a sampling distribution. If all the possible samples of a given size are taken from a population and their variances are calculated, the probability distribution of these variances is called the sampling distribution of the sample variance.

The random variable

follows a Chi–square distribution with degrees of freedom.

To find the formula for calculating Confidence intervals for the variance, it is necessary to introduce new notations.

We will denote the number for which (Fig.6.13)

Similarly, it follows that is defined as

And is defined as .

Then it follows that

In the end, as it shown in Fig.6.14

Using usual procedure we obtain confidence interval for population variance as

where follows a Chi – square distribution with degrees of freedom.

Example:

The variance in drug weights is very critical in the pharmaceutical industry. For a specific drug, with weights measured in grams, a sample of 18 units provided a sample variance of

a) Construct a 90 % confidence interval estimate for the population variance for the weights of this drug.

b) Construct a 90 % confidence interval estimate for the population standard deviation for the weights of this drug.

Solution:

From the given information

; ;

and for a 90 % confidence interval, ; .

confidence interval for is given by

a) From the Chi- square table we obtain that

;

After substitution we obtain

b) We can obtain the confidence interval for the population standard deviation by taking the positive square root of the two limits of the above confidence interval for the population variance. Thus, 90 % confidence interval estimate for the population standard deviation is

Hence, the standard deviation of all investigated drug is between 0.47 and 0.84 grams at a 90 % confidence level.

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