State suitable hypotheses to test the inspector’s claim.

Find unbiased estimates of $\mu$ and ${\sigma ^2}$.

Carry out an appropriate test and state the $p$-value obtained.

Using a 10% significance level and justifying your answer, state your conclusion in context.

${H_0}:\mu = 7,{\text{ }}{H_1}:\mu < 7$     A1

[1 mark]

$\bar x = \frac{{83.64}}{{12}} = 6.97$     A1

$s_{n - 1}^2 = \frac{{583.05}}{{11}} - \frac{{{\text{ }}{{83.64}^2}}}{{132}} = 0.0072$     (M1)A1

[3 marks]

$t = \frac{{6.97 - 7}}{{\sqrt {\frac{{0.0072}}{{12}}} }} = - 1.22(474 \ldots )$     (M1)(A1)

${\text{degrees of freedom}} = 11$     (A1)

$p{\text{ - value}} = 0.123$     A1

Note:     Accept any answer that rounds correctly to 0.12.

[4 marks]

because $p > 0.1$     R1

the inspector’s claim is not supported (at the 10% level)

(or equivalent in context)     A1

Note:     Only award the A1 if the R1 has been awarded

[2 marks]