A car manufacturer claims that the front tyres on a particular model of car have an average life span of 20000 miles. Following comments from customers, the sales manager wishes to test if the hife span of the tyres is greater than 20000 miles.

(a) Explain why the sales manager should carry out a 1-tail test. State hypotheses for the test, defining any symbols you use.

[3]

The sales manager contacts customers and gathers details about the life spans of a random sample of 50 of these tyres. The life spans, $x$ thousand miles, are summarised below.

$\Sigma(x-20)=9.4 \quad \Sigma(x-20)^2=38.76$

(b) Calculate unbiased estimates of the population mean and variance of the life spans of the tyres.

[2]

(c) Test, at the $5 \%$ level of significance, whether the mean life span of front tyres is more than 20000 miles.

[3]

(d) Explain why this test would be inappropriate if the sales manager had taken a random sample of 15 tyres.

[1]

$20188, 755000 ; p=0.0630$

[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.