Further Statistics 2: Other hypothesis tests and confidence intervalsEdexcel A-Level Further Maths: Topic test

1. 1
   A temperature sensor gives readings, in $^{\circ}$C, that are Normally distributed. The manufacturer claims that the population variance is $0.25$. A random sample of $16$ readings of a fixed temperature has sample variance $s^2=0.384$. A test is to be carried out of $\mathrm{H}_0:\sigma^2=0.25$ against $\mathrm{H}_1:\sigma^2>0.25$.

   (a)

   How many degrees of freedom does the $\chi^2$ test statistic have?

   [1 mark]

   • A$16$
   • B$15$
   • C$14$
   • D$30$

   (b)

   What is the value of the test statistic?

   [1 mark]

   • A$1.536$
   • B$24.58$
   • C$0.651$
   • D$23.04$

   (c)

   The critical value of $\chi^2_{15}$ for a one-tailed test at the $5\%$ level is $24.996$. Complete the test, stating your conclusion in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two laboratories measure the concentration of a pollutant in river samples, and the results from each are Normally distributed. Independent random samples give $n_1=11$ with $s_1^2=18.5$ from laboratory $1$, and $n_2=9$ with $s_2^2=7.4$ from laboratory $2$. A test is carried out of $\mathrm{H}_0:\sigma_1^2=\sigma_2^2$ against $\mathrm{H}_1:\sigma_1^2\neq\sigma_2^2$.

   (a)

   What is the value of the test statistic $\frac{s_1^2}{s_2^2}$?

   [1 mark]

   • A$2.5$
   • B$0.4$
   • C$11.1$
   • D$1.58$

   (b)

   What are the degrees of freedom of the $F$ distribution for $\frac{S_1^2}{S_2^2}$ under $\mathrm{H}_0$?

   [1 mark]

   • A$(11,9)$
   • B$(8,10)$
   • C$(10,8)$
   • D$(10,9)$

   (c)

   The critical value of $F_{10,8}$ for the upper $5\%$ tail is $3.347$. Complete the test at the $10\%$ significance level, stating your conclusion in context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The thickness of ceramic tiles, in mm, is Normally distributed. A random sample of $12$ tiles gives $\sum x=300.6$ and $\sum x^2=7531.13$. The specification says that the variance of the thickness should be $0.06$.

   (a)

   Calculate the unbiased estimate of the population variance.

   [3 marks]

   (b)

   Test, at the $5\%$ significance level, whether the variance of the thickness exceeds the specified value. The critical value of $\chi^2_{11}$ for the upper $5\%$ tail is $19.675$. State your hypotheses and your conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A pharmacist compares the time, in minutes, for a tablet to dissolve in two formulations, $A$ and $B$. The dissolving times are Normally distributed. A random sample of $15$ tablets of formulation $A$ has sample variance $0.84$ and an independent random sample of $11$ tablets of formulation $B$ has sample variance $0.35$.

   (a)

   Find a $95\%$ confidence interval for the population variance of the dissolving time for formulation $A$. For $\chi^2_{14}$, the lower $2.5\%$ point is $5.629$ and the upper $2.5\%$ point is $26.119$.

   [6 marks]

   (b)

   Test, at the $5\%$ significance level, whether the variance of the dissolving time is greater for formulation $A$ than for formulation $B$. The upper $5\%$ point of $F_{14,10}$ is $2.865$. State your hypotheses, the assumptions needed and your conclusion in context.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A $90\%$ confidence interval is to be found for the variance $\sigma^2$ of a Normal population from a random sample of $21$ observations with sample variance $s^2=6.4$. For $\chi^2_{20}$, the lower $5\%$ point is $10.851$ and the upper $5\%$ point is $31.410$.

   (a)

   What is the lower limit of the confidence interval for $\sigma^2$?

   [1 mark]

   • A$4.28$
   • B$11.80$
   • C$4.08$
   • D$2.02$

   (b)

   What is the corresponding $90\%$ confidence interval for the standard deviation $\sigma$?

   [1 mark]

   • A$(2.02,\ 3.43)$
   • B$(4.08,\ 11.80)$
   • C$(2.07,\ 3.52)$
   • D$(2.02,\ 11.80)$

   (c)

   Use the interval for $\sigma^2$ to decide whether $\mathrm{H}_0:\sigma^2=12$ would be rejected in a two-tailed test at the $10\%$ level, and state the assumption on which the interval depends.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Two independent random samples are taken from Normal populations. Sample $1$ has $n_1=8$ and $s_1^2=5.6$. Sample $2$ has $n_2=13$ and $s_2^2=2.0$. A test is carried out of $\mathrm{H}_0:\sigma_1^2=\sigma_2^2$ against $\mathrm{H}_1:\sigma_1^2>\sigma_2^2$, using the test statistic $\frac{S_1^2}{S_2^2}\sim F_{7,12}$ under $\mathrm{H}_0$.

   (a)

   Which assumption is needed for this test to be valid?

   [1 mark]

   • AThe population means are equal
   • BThe sample sizes are equal
   • CThe sample variances are equal
   • DBoth populations are Normally distributed and the samples are independent

   (b)

   The upper $5\%$ point of $F_{7,12}$ is $2.913$. What is the conclusion at the $5\%$ level?

   [1 mark]

   • A$F=2.8<2.913$, so reject $\mathrm{H}_0$
   • B$F=2.8<2.913$, so do not reject $\mathrm{H}_0$
   • C$F=0.357<2.913$, so do not reject $\mathrm{H}_0$
   • D$F=3.6>2.913$, so reject $\mathrm{H}_0$

   (c)

   The upper $10\%$ point of $F_{7,12}$ is $2.283$. State the conclusion of the test at the $10\%$ significance level, and comment on your result compared with the $5\%$ level.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A watch manufacturer states that the daily error, in seconds, of its watches is Normally distributed with variance $4$. A random sample of $25$ watches has sample variance $s^2=2.3$. A test is carried out of $\mathrm{H}_0:\sigma^2=4$ against $\mathrm{H}_1:\sigma^2\neq4$.

   (a)

   Find the value of the test statistic and state its distribution under $\mathrm{H}_0$.

   [3 marks]

   (b)

   For $\chi^2_{24}$, the lower $2.5\%$ point is $12.401$ and the upper $2.5\%$ point is $39.364$. Complete the test at the $5\%$ significance level, stating your conclusion in context.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A lens manufacturer has two production lines. The thickness of a lens, in mm, from each line is Normally distributed. Independent random samples give $n_L=13$ with sample variance $0.0084$ from line $L$, and $n_M=16$ with sample variance $0.0036$ from line $M$.

   (a)

   Test, at the $10\%$ significance level, whether the variances of the thickness of lenses from the two lines are different. The upper $5\%$ point of $F_{12,15}$ is $2.475$. State your conclusion in context.

   [6 marks]

   (b)

   Find a $95\%$ confidence interval for the variance of the thickness of lenses from line $L$. For $\chi^2_{12}$, the lower $2.5\%$ point is $4.404$ and the upper $2.5\%$ point is $23.337$. The target variance is $0.005$. Comment on the target, and explain why the sample variance for line $M$ lying below your interval does not contradict part (a).

   [6 marks]

   Total for question 8: 12 marks

End of questions