Plastic sheets produced by a machine are periodically...

Plastic sheets produced by a machine are periodically monitored for possible fluctuations in thickness. If the true variance in thicknesses exceeds 2.25 square millimeters, there is cause for concern about product quality. Thickness measurements for a random sample of 10 sheets produced in a particular shift were taken, giving the following results (in millimeters): $\begin{array}{llllllll}226 & 226 & 232 & 227 & 225 & 228 & 225 & 228\end{array}$ $229 \quad 230$ a. Find the sample variance. b. Test, at the $5 \%$ significance level, the null hypothesis that the population variance is at most 2.25 .

Plastic sheets produced by a machine are periodically monitored for possible fluctuations in thickness. If the true variance in thicknesses exceeds 2.25 square millimeters, there is cause for concern about product quality. Thickness measurements for a random sample of 10 sheets produced in a particular shift were taken, giving the following results (in millimeters):
$\begin{array}{llllllll}226 & 226 & 232 & 227 & 225 & 228 & 225 & 228\end{array}$
$229 \quad 230$
a. Find the sample variance.
b. Test, at the $5 \%$ significance level, the null hypothesis that the population variance is at most 2.25 .

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Key Concepts

Hypothesis Testing for Variance

This involves setting up null and alternative hypotheses regarding the population variance and using the chi-square test to determine whether the observed sample variance provides enough evidence to reject the null hypothesis at a chosen significance level.

Chi-Square Distribution

The chi-square distribution is used in inferential statistics to test hypotheses about the variance of a normally distributed population. The test statistic is computed by multiplying the sample variance by the degrees of freedom and dividing by the hypothesized variance.

Degrees of Freedom

Degrees of freedom, in this context n - 1, represent the number of values in the final calculation of a statistic that are free to vary. They are crucial for obtaining the correct distribution for the test statistic in variance analysis.

Sample Variance

The sample variance is a measure of the spread of the data around the sample mean, calculated as the sum of squared deviations from the mean divided by the degrees of freedom (n - 1) to ensure unbiased estimation of the population variance.

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