t tests for a mean and the paired t-testEdexcel A-Level Further Maths: Subtopic test

1. 1
   The mass of a chocolate bar is Normally distributed. The label states a mean mass of $50$ g. A random sample of $10$ bars has sample mean $49.2$ g and sample standard deviation $s=2.1$ g. A test is carried out to see whether the mean mass is less than the label states.

   (a)

   Under $H_0:\mu=50$, which distribution does $\frac{\bar X-\mu}{S/\sqrt n}$ have?

   [1 mark]

   • A$t_9$
   • B$t_{10}$
   • C$\mathrm{N}(0,1)$
   • D$\chi^2_9$

   (b)

   Calculate the test statistic.

   [1 mark]

   • A$-0.38$
   • B$-1.14$
   • C$-3.81$
   • D$-1.20$

   (c)

   The lower $5\%$ point of $t_9$ is $-1.833$. 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 $8$ observations from a Normal population has sample mean $24.3$ and sample standard deviation $s=1.9$. The population variance is unknown.

   (a)

   Which critical value is used for a $95\%$ confidence interval for the mean?

   [1 mark]

   • A$1.960$
   • B$2.365$
   • C$2.306$
   • D$1.895$

   (b)

   Find the $95\%$ confidence interval for the population mean.

   [1 mark]

   • A$(22.98,\ 25.62)$
   • B$(19.81,\ 28.79)$
   • C$(22.71,\ 25.89)$
   • D$(22.75,\ 25.85)$

   (c)

   Explain why the $t$ distribution is used rather than the Normal distribution, and state the assumption needed about the population.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A supplier claims that the mean length of its rods is $12.0$ cm. The lengths are Normally distributed. A random sample of $8$ rods has $\sum x=99.4$ and $\sum x^2=1236.48$, where $x$ is the length in cm.

   (a)

   Calculate the sample mean and an unbiased estimate of the population variance.

   [3 marks]

   (b)

   Test, at the $5\%$ significance level, whether the supplier's claim is correct. The upper $2.5\%$ point of $t_7$ is $2.365$. State your conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Eight students sat a test before and after a revision course. Scores before: $52,\ 61,\ 47,\ 58,\ 66,\ 49,\ 55,\ 63$. Scores after, in the same order of students: $57,\ 64,\ 46,\ 65,\ 71,\ 55,\ 58,\ 70$. The differences (after $-$ before) may be assumed to be Normally distributed. Let $\mu_d$ be the population mean difference.

   (a)

   Test, at the $5\%$ significance level, whether the revision course increases mean scores. The upper $5\%$ point of $t_7$ is $1.895$. State your hypotheses and conclusion.

   [6 marks]

   (b)

   (i) Explain why a paired $t$-test is appropriate here, and state the assumption required.
   (ii) Using $s_d=2.669$, find the smallest mean difference $\bar d$ that would give a significant result at the $5\%$ level.
   (iii) Evaluate whether the result shows that the course caused the improvement.

   [6 marks]

   Total for question 4: 12 marks

End of questions