**Hypothesis Testing: Testing for a population variance**

A hypothesis testing is a procedure in which a claim about a certain population parameter is tested. A population parameter is a numerical constant that represents o characterizes a distribution. Typically, a hypothesis test is about a population mean, typically notated as \(\mu\), but in reality it can be about any population parameter, such a population proportion \(p\), or a population standard deviation \(\sigma\).

In this case, we are going to analyze the case of a hypothesis test involving a population standard deviation \(\sigma\). As with any type of hypothesis testing, sample data is required to test a claim about \(\sigma\). Notice that sometimes the claim involves the population variance \({{\sigma }^{2}}\) instead, but it is essentially the same thing because, for example, making the claim about the population variance that \({{\sigma }^{2}}=16\) is absolutely equivalent to making the claim \(\sigma =4\) about the population standard deviation. So therefore, always keep in mind that making a claim about the population variance has always paired a claim about the population standard deviation, and vice versa.

The procedures for determining the null and alternative hypotheses and the type of tail for the test are applied all the same the steps used for testing a claim about the population mean (This is, we state the given claim(s) in mathematical form and examine the type of sign involved).

__EXAMPLE__

Assume that an official from the treasury claims that post-1983 pennies have weights with a standard deviation greater than 0.0230 g. Assume that a simple random sample of n = 25 pre-1983 pennies is collected, and that sample has a standard deviation of 0.03910 g. Use a 0.05 significance level to test the claim that pre-1983 pennies have weights with a standard deviation greater than 0.0230 g. Based on these sample results, does it appear that weights of pre-1983 pennies vary more than those of post-1983 pennies?

__HOW DO WE SOLVE THIS?__

We need to test

\[\begin{align}{H}_{0}: \sigma \le {0.0230} \\ {{H}_{A}}: \sigma > {0.0230} \\ \end{align}\]

The value of the Chi Square statistics is computed as

\[{{\chi }^{2}}=\frac{\left( n-1 \right){{s}^{2}}}{{{\sigma }^{2}}}=\frac{\left( 25-1 \right)\times {0.03910^2}}{0.0230^2}= {69.36}\]

The upper critical value for \(\alpha = 0.05\) and *df* = 24 is

\[\chi _{upper}^{2}= {36.415}\]

which means that we reject the null hypothesis.

*This means that we have enough evidence to support the claim that weights of pre-1983 pennies vary more than those of post-1983 pennies, at the 0.05 significance level. *

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