IBDP MAI : Topic 4 Statistics and probability - AHL 4.15 A linear combination of n independent normal AI HL Paper 3
Question
A random variable $X$ has a distribution with mean $\mu$ and variance 4. A random sample of size 100 is to be taken from the distribution of $X$.
Josie takes a different random sample of size 100 to test the null hypothesis that $\mu=60$ against the alternative hypothesis that $\mu>60$ at the $5 \%$ level.
a. State the central limit theorem as applied to a random sample of size $n$, taken from a distribution with mean $\mu$ and variance $\sigma^2$.
b. Jack takes a random sample of size 100 and calculates that $\bar{x}=60.2$. Find an approximate $90 \%$ confidence interval for $\mu$.
c.i. Find the critical region for Josie’s test, giving your answer correct to two decimal places.
c.ii.Write down the probability that Josie makes a Type I error.
c.iiiGiven 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.
▶️Answer/Explanation
a. for $n$ (sufficiently) large the sample mean $\bar{X}$ approximately
A1
$\sim \mathrm{N}\left(\mu, \frac{\sigma^2}{n}\right)$
A1
Note: Award the first $\boldsymbol{A 1}$ for $n$ large and reference to the sample mean $(\bar{X})$, the second $\boldsymbol{A 1}$ is for normal and the two parameters.
Note: Award the second $\boldsymbol{A 1}$ only if the first $\boldsymbol{A 1}$ is awarded.
Note: Allow ‘ $n$ tends to infinity’ or ‘ $n \geq 30$ ‘ in place of ‘large’.
[2 marks]
b. $[59.9,60.5]$
A1A1
Note: Accept answers which round to the correct 3sf answers.
[2 marks]
c.i. under $H_0, \bar{X} \sim \mathrm{N}\left(60, \frac{4}{100}\right)$
(A1)
required to find $k$ such that $P(\bar{X}>k)=0.05$
(M1)
use of any valid method, eg GDC Inv(Normal) or $k=60+z \frac{\sigma}{\sqrt{n}}$
(M1)
hence critical region is $\bar{x}=60.33$
A1
[4 marks]
c.ii. 0.05
A1
[1 mark]
c.iiiP(Type II error) $=\mathrm{P}\left(H_0\right.$ is accepted $/ H_0$ is false $)$
(R1)
Note: Accept Type II error means $H_0$ is accepted given $H_0$ is false.
$$
\begin{aligned}
& \Rightarrow \mathrm{P}(\bar{X}<60.33)=0.25 \text { when } \bar{X} \sim \mathrm{N}\left(\mu, \frac{4}{100}\right) \\
& \Rightarrow \mathrm{P}\left(\frac{\bar{X}-\mu}{\frac{2}{10}}<\frac{60.33-\mu}{\frac{2}{10}}\right)=0.25 \quad \text { (M1) } \\
& \Rightarrow \mathrm{P}\left(Z<\frac{60.33-\mu}{\frac{2}{10}}\right)=0.25 \text { where } Z \sim \mathrm{N}\left(0,1^2\right) \\
& \frac{60.33-\mu}{\frac{2}{10}}=-0.6744 \ldots \\
& \mu=60.33+\frac{2}{10} \times 0.6744 \ldots \\
& \mu=60.5 \\
& {[5 \text { marks }]}
\end{aligned}
$$
(M1)
A1
[5 marks]