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

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

1. 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
2. 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
3. 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