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Cambridge International AS and A Level

Name of student HENRYTAIGO Date
Adm. number Year/grade HenryTaigo Stream HenryTaigo
Subject Probability & Statistics 2 (S2) Variant(s) P71, P72, P73
Start time Duration Stop time

Qtn No. 1 2 3 4 Total
Marks 10 14 10 10 44

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Question 1 Code: 9709/71/M/J/13/7, Topic: -

Leila suspects that a particular six-sided die is biased so that the probability, $p$, that it will show a six is greater than $\frac{1}{6}$. She tests the die by throwing it 5 times. If it shows a six on 3 or more throws she will conclude that it is biased.

$\text{(i)}$ State what is meant by a Type I error in this situation and calculate the probability of a Type I error. $[3]$

$\text{(ii)}$ Assuming that the value of $p$ is actually $\frac{2}{3}$, calculate the probability of a Type II error. $[3]$

Leila now throws the die 80 times and it shows a six on 50 throws.

$\text{(iii)}$ Calculate an approximate $96 \%$ confidence interval for $p$. $[4]$

Question 2 Code: 9709/73/M/J/13/7, Topic: -

In the past the weekly profit at a store had mean $\$ 34600$ and standard deviation $\$ 4500$. Following a change of ownership, the mean weekly profit for 90 randomly chosen weeks was $\$ 35400$.

$\text{(i)}$ Stating a necessary assumption, test at the $5 \%$ significance level whether the mean weekly profit has increased. $[6]$

$\text{(ii)}$ State, with a reason, whether it was necessary to use the Central Limit theorem in part $\text{(i)}$. $[2]$

The mean weekly profit for another random sample of 90 weeks is found and the same test is carried out at the $5 \%$ significance level.

$\text{(iii)}$ State the probability of a Type I error. $[1]$

$\text{(iv)}$ Given that the population mean weekly profit is now $\$ 36500$, calculate the probability of a Type II error. $[5]$

Question 3 Code: 9709/71/O/N/13/7, Topic: -

Kieran and Andreas are long-jumpers. They model the lengths, in metres, that they jump by the independent random variables $K \sim \mathrm{N}(5.64,0.0576)$ and $A \sim \mathrm{N}(4.97,0.0441)$ respectively. They each make a jump and measure the length. Find the probability that

$\text{(i)}$ the sum of the lengths of their jumps is less than $11 \mathrm{~m}$, $[4]$

$\text{(ii)}$ Kieran jumps more than $1.2$ times as far as Andreas. $[6]$

Question 4 Code: 9709/72/O/N/13/7, Topic: -

Worked solutions: P1, P3 & P6 (S1)

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