Shape of the chi-squared distribution?

Positively skewed, only positive values.

For H1:\sigma^2\sigma0^2, when do you reject H0?

When the statistic exceeds the upper critical value of \chi^2{n-1}.

Confidence interval for \sigma^2?

\left(\frac{(n-1)s^2}{\chi^2{upper}},\ \frac{(n-1)s^2}{\chi^2{lower}}\right)

Which chi-squared point gives the lower limit of the interval?

The upper (larger) point, because it is in the denominator.

Distribution used in the test of two variances?

\frac{S1^2}{S2^2}\sim F{n1-1,\,n2-1} under H0

For a two-tailed F-test at \alpha\%, which critical value is used?

The upper \frac{\alpha}{2}\% point, with the larger variance on top.

Assumptions for the F-test?

Independent random samples from Normal populations.

A 95\% interval for \sigma^2 corresponds to which test?

A two-tailed test at the 5\% level.

Variance of a Normal distribution and the F-testEdexcel A-Level Further Maths: Flashcards

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Distribution of $\frac{(n-1)S^2}{\sigma^2}$?

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Distribution of $\frac{(n-1)S^2}{\sigma^2}$ ?: $\chi^2_{n-1}$ , for a sample from a Normal population.
Degrees of freedom for a variance test with a sample of size $n$ ?: $n-1$
Test statistic for $H_0:\sigma^2=\sigma_0^2$ ?: $\frac{(n-1)s^2}{\sigma_0^2}$
Shape of the chi-squared distribution?: Positively skewed, only positive values.
For $H_1:\sigma^2>\sigma_0^2$ , when do you reject $H_0$ ?: When the statistic exceeds the upper critical value of $\chi^2_{n-1}$ .
Confidence interval for $\sigma^2$ ?: $\left(\frac{(n-1)s^2}{\chi^2_{\text{upper}}},\ \frac{(n-1)s^2}{\chi^2_{\text{lower}}}\right)$
Which chi-squared point gives the lower limit of the interval?: The upper (larger) point, because it is in the denominator.
Is the confidence interval for $\sigma^2$ symmetrical about $s^2$ ?: No, because $\chi^2$ is skewed.
Distribution used in the test of two variances?: $\frac{S_1^2}{S_2^2}\sim F_{n_1-1,\,n_2-1}$ under $H_0$
Test statistic for the F-test?: $F=\frac{s_1^2}{s_2^2}$
For a two-tailed F-test at $\alpha\%$ , which critical value is used?: The upper $\frac{\alpha}{2}\%$ point, with the larger variance on top.
Assumptions for the F-test?: Independent random samples from Normal populations.
A $95\%$ interval for $\sigma^2$ corresponds to which test?: A two-tailed test at the $5\%$ level.

Exam questions on Variance of a Normal distribution and the F-test

The lengths of bolts produced by a machine are Normally distributed. The manufacturer states that the population variance is $1.6$ mm $^2$ . A random sample of $16$ bolts has sample variance $s^2=2.5$ mm $^2$ . A test is carried out to see whether the variance is greater than stated.
The upper $5\%$ point of $\chi^2_{15}$ is $24.996$ . Complete the test at the $5\%$ significance level and state your conclusion in context.2 marks
A random sample of $10$ observations from a Normal population has sample variance $s^2=8.4$ . A $90\%$ confidence interval for the population variance $\sigma^2$ is to be found using the $\chi^2$ distribution.
A claim is made that $\sigma^2=25$ . Use your interval to comment on this claim, stating the significance level of the corresponding test.2 marks
Independent random samples are taken from two Normal populations. Sample $1$ : $n_1=12$ , $s_1^2=14.4$ . Sample $2$ : $n_2=9$ , $s_2^2=5.6$ . A test is carried out of whether the two populations have equal variances.
State suitable hypotheses and calculate the test statistic.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).