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9
(i)
From the graphing calculator,

unbiased estimate of $\mu = 12.8$

unbiased estimate of ${\sigma}^{2} = 2.31$

(ii)
We assume that 2.31 is a good estimate of the unknown population variance.
We assume the distance travelled per litre of fuel by a car is independent of another car.

$H_0: \mu = 13.8$
$H_1: \mu < 13.8$ Under $H_0$, perform a left-tailed T-test.

Test statistic, $T ~ = ~ \frac{\bar{X}-\mu}{\frac{s}{\sqrt{n}}} ~ \sim ~ t(7)$

Using the graphing calculator, p-value $= 0.052397 > 0.05$, thus we do not reject $H_0$.

There is insufficient evidence at 5% significance level to say that the distance travelled per litre of fuel is too high.

This is a fairly manageable question that should be attempted. The only issue is probably with the assumption part. Students cannot use their regular “assume population mean followed a normal distribution” assumption since the question stated it. So they need to be more creative. As to what constitutes a good estimate, we consider if they are consistent and unbiased.

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