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

1. 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.

   (a)

   Under $H_0:\sigma^2=1.6$, which distribution does $\frac{(n-1)S^2}{\sigma^2}$ follow?

   [1 mark]

   • A$\chi^2_{16}$
   • B$\chi^2_{15}$
   • C$t_{15}$
   • D$\mathrm{N}(0,1)$

   (b)

   Calculate the value of the test statistic.

   [1 mark]

   • A$23.4$
   • B$25.0$
   • C$1.56$
   • D$4.0$

   (c)

   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]

   Total for question 1: 4 marks

2. 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)

   Which critical values of $\chi^2_9$ are needed to find the $90\%$ confidence interval?

   [1 mark]

   • A$2.700$ and $19.023$
   • B$3.940$ and $18.307$
   • C$3.325$ and $16.919$
   • D$3.325$ and $19.023$

   (b)

   Find the $90\%$ confidence interval for $\sigma^2$.

   [1 mark]

   • A$(3.10,\ 15.79)$
   • B$(3.97,\ 28.0)$
   • C$(4.13,\ 19.19)$
   • D$(4.47,\ 22.7)$

   (c)

   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]

   Total for question 2: 4 marks

3. 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.

   (a)

   State suitable hypotheses and calculate the test statistic.

   [3 marks]

   (b)

   The upper $5\%$ point of $F_{11,8}$ is $3.313$. Carry out the test at the $10\%$ significance level, and state your conclusion and one assumption you have made.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A machine fills bottles with a volume of liquid, in ml, that is Normally distributed. The variance is supposed to be $4$ ml$^2$. A random sample of $20$ bottles has sample variance $6.8$ ml$^2$.

   (a)

   Test, at the $5\%$ level, whether the variance of the volume is greater than $4$ ml$^2$. The upper $5\%$ point of $\chi^2_{19}$ is $30.144$. State your hypotheses and conclusion in context.

   [6 marks]

   (b)

   (i) The upper and lower $2.5\%$ points of $\chi^2_{19}$ are $32.852$ and $8.907$. Find a $95\%$ confidence interval for $\sigma^2$.
   (ii) The claim $\sigma^2=4$ lies inside your interval, yet it was rejected in part (a). Explain why these results do not contradict each other.

   [6 marks]

   Total for question 4: 12 marks

End of questions