The mean weight of a certain breed of bird is believed to be 2.5 kg. In order to test this belief, it is planned to determine the weights ${x_1}{\text{ , }}{x_2}{\text{ , }}{x_3}{\text{ , }} \ldots {\text{, }}{x_{16}}$ (in kg) of sixteen of these birds and then to calculate the sample mean ${\bar x}$ . You may assume that these weights are a random sample from a normal distribution with standard deviation 0.1 kg.

(a)     State suitable hypotheses for a two-tailed test.

(b)     Find the critical region for ${\bar x}$ having a significance level of 5 %.

(c)     Given that the mean weight of birds of this breed is actually 2.6 kg, find the probability of making a Type II error.

(a)     ${H_0}:\mu = 2.5$     A1

${H_1}:\mu \ne 2.5$     A1

[2 marks]

(b)     the critical values are $2.5 \pm 1.96 \times \frac{{0.1}}{{\sqrt {16} }}$ ,     (M1)(A1)(A1)

i.e. 2.45, 2.55     (A1)

the critical region is $\bar x < 2.45 \cup \bar x > 2.55$     A1A1

Note: Accept $\leqslant ,{\text{ }} \geqslant$ .

[6 marks]

(c)     ${\bar X}$ is now ${\text{N}}(2.6,{\text{ }}{0.025^2})$     A1

a Type II error is accepting ${H_0}$ when ${H_1}$ is true     (R1)

thus we require

${\text{P}}(2.45 < \bar X < 2.55)$     M1A1

$= 0.0228\,\,\,\,\,$(Accept 0.0227)     A1

Note: If critical values of 2.451 and 2.549 are used, accept 0.0207.

[5 marks]

Total [13 marks]

In (a), some candidates incorrectly gave the hypotheses in terms of ${\bar x}$ instead of $\mu$. In (b), many candidates found the correct critical values but then some gave the critical region as $2.45 < \bar x < 2.55$ instead of $\bar x < 2.45 \cup \bar x > 2.55$ Many candidates gave the critical values correct to four significant figures and therefore were given an arithmetic penalty. In (c), many candidates correctly defined a Type II error but were unable to calculate the corresponding probability.