A manufacturer has been purchasing raw materials from a...

A manufacturer has been purchasing raw materials from a supplier whose consignments have a variance of 15.4 (in squared pounds) in impurity levels. A rival supplier claims that she can supply consignments of this raw material with the same mean impurity level but with lower variance. For a random sample of 25 consignments from the second supplier, the variance in impurity levels was found to be 12.2 . What is the probability of observing a value this low or lower for the sample variance if, in fact, the true population variance is 15.4? Assume that the population distribution is normal.

Key Concepts

Sampling Distribution of the Sample Variance

When sampling from a normally distributed population, the sample variance does not have a normal distribution. Instead, if you scale the sample variance by the true population variance and multiply by the degrees of freedom (n – 1), the resulting statistic follows a chi?square distribution. This concept underpins tests concerning the variability in the data.

Chi?Square Distribution

The chi?square distribution arises as the distribution of a sum of squared independent standard normal random variables. In the context of variance estimation, the statistic (n – 1)S²/?² is chi?square distributed with (n – 1) degrees of freedom. This provides the basis for constructing confidence intervals and hypothesis tests for a population variance.

P-Value in Hypothesis Testing for Variances

In tests concerning population variance, the P-value represents the probability of obtaining a sample variance as extreme as or more extreme than the one observed under the null hypothesis. It is computed by comparing the calculated chi?square statistic to the chi?square distribution to determine the likelihood of such an observation if the claimed population variance were correct.

Degrees of Freedom

Degrees of freedom, typically n – 1 for variance estimation, represent the number of independent values that are free to vary in a statistical calculation. They are crucial in determining the exact form of the chi?square distribution used in variance tests, affecting both the shape and spread of the distribution.

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