(a)     Calculate an unbiased estimate for

  (i)     \(\mu \) ,

  (ii)     \({\sigma ^2}\) .

(b)     The value of \(\mu \) is thought to be 4.5, so the following hypotheses are defined.\[{{\text{H}}_0}:\mu  = 4.5;{\text{ }}{{\text{H}}_1}:\mu  < 4.5\]

  (i)     Find the p-value of the observed sample mean.

  (ii)     State your conclusion if the significance level is

    (a)     1 %,

    (b)     10 %.

(a)     (i)     \(\bar x = \frac{{4.35 + 4.53}}{2} = 4.44\) (estimate of \(\mu \))     A2

(ii)     Degrees of freedom = 9     (A1)

Critical value of t = 2.262     (A1)

\(2.262 \times \frac{s}{{\sqrt {10} }} = 0.09\)     M1A1

\(s = 0.12582…\)     (A1)

\({s^2} = 0.0158\) (estimate of \({\sigma ^2}\))     A1

[8 marks]

(b)     (i)     Using t test     (M1)

\(t = \frac{{4.44 - 4.5}}{{\sqrt {\frac{{0.0158}}{{10}}} }} = - 1.50800\)   (Accept \( - 1.50946\))     (A1)

p-value = 0.0829 (Accept 0.0827)     A2

(ii)     (a)     Accept \({{\text{H}}_0}\) / Reject \({{\text{H}}_1}\) .     R1

  (b)     Reject \({{\text{H}}_0}\) / Accept \({{\text{H}}_1}\) .     R1

[6 marks]

Total [14 marks]

Most candidates realised that the unbiased estimate of the mean was simply the central point of the confidence interval. Many candidates, however, failed to realise that, because the variance was unknown, the t-distribution was used to determine the confidence limits. In (b), although the p-value was asked for specifically, some candidates solved the problem correctly by comparing the value of their statistic with the appropriate critical values. This method was given full credit but, of course, marks were lost by their failure to give the p-value.