$20188, 755000 ; p=0.0630$

[Maximum marks: 9 marks]
(a) Manager should carry out a 1-tail test since he wishes to test if the population mean life span is greater than 20000 miles.
Let $\mu$ be the population mean life span of tyres.
Let $H_0$ be the null hypothesis.
Let $H_1$ be the alternative hypothesis.
Test $H_0: \mu=20000$
against $H_1: \mu>20000$

(b)
Unbiased estimates of population mean $=\left(\frac{9.4}{50}+20\right) \times 1000=20188$ miles
Unbiased estimates of population variance
$=\left[\frac{1}{49}\left(38.76-\frac{9.4^2}{50}\right)\right] \times 1000^2=754955.102 \approx 755000 \mathrm{miles}^2$
Note that if you did not multiply, i.e., did in thousand miles. It is perfectly fine, but you should note to state the $H_0: \mu=20$ instead, in order to be coherent for the $p$ value.

(c)
Under $H_0$, since $n$ is sufficiently large, by central limit theorem,
$\bar{X} \sim \mathrm{N}\left(20000, \frac{754955.102}{50}\right)$ approximately.
$p=\mathrm{P}(\bar{X}>20188)=0.06301$
Since $p \approx 0.0630>0.05$, we do not reject $H_0$ at $5 \%$ significance level and conclude with insufficient evidence that the mean life span of the tyre is more than 20000 miles.

(d)
As the distribution of the tyre is unknown initially, we will need to take a sufficiently large sample size that is greater than 30 in order to approximate the sample mean life span to a normal distribution by central limit theorem. Thus a sample size of 15 will be inappropriate.