State the central limit theorem as applied to a random sample of size $n$, taken from a distribution with mean $\mu$ and variance ${\sigma }^2$.

Jack takes a random sample of size 100 and calculates that $\overline{x}=60.2$. Find an approximate 90 % confidence interval for $\mu$.

Find the critical region for Josie’s test, giving your answer correct to two decimal places.

Write down the probability that Josie makes a Type I error.

Given that the probability that Josie makes a Type II error is 0.25, find the value of $\mu$, giving your answer correct to three significant figures.

for $n$ (sufficiently) large the sample mean $\overline{X}$ approximately           A1

$\sim \text{N}\left(\mu \text{, }\frac{{\sigma }^2}{n}\right)$           A1

Note: Award the first A1 for $n$ large and reference to the sample mean $\left(\overline{X}\right)$, the second A1 is for normal and the two parameters.

Note: Award the second A1 only if the first A1 is awarded.

Note: Allow ‘$n$ tends to infinity’ or ‘$n$ ≥ 30’ in place of ‘large’.

[2 marks]

[59.9, 60.5]                   A1A1

Note: Accept answers which round to the correct 3sf answers.

[2 marks]

under $H_0$, $\overline{X}\sim \text{N}\left(60\text{, }\frac{4}{100}\right)$                   (A1)

required to find $k$ such that                    (M1)

use of any valid method, eg GDC Inv(Normal) or $k=60+z\frac{\sigma }{\sqrt{n}}$                   (M1)

hence critical region is $\overline{x}=60.33$                   A1

[4 marks]

0.05                   A1

[1 mark]

$\text{P}$(Type II error) = $\text{P}$($H_0$ is accepted / $H_0$ is false)       (R1)

Note: Accept Type II error means $H_0$ is accepted given $H_0$ is false.

when $\overline{X}\sim \text{N}\left(\mu \text{, }\frac{4}{100}\right)$       (M1)

       (M1)

  where  $Z\sim \text{N}\left(0\text{, }{1}^2\right)$

$\frac{60.33-\mu }{\frac{2}{10}}=-0.6744\dots$       (A1)

$\mu =60.33+\frac{2}{10}\times 0.6744\dots$

$\mu =60.5$                   A1

[5 marks]