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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 5 6 Total
Marks 6 7 6 5 5 7 36

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Question 1 Code: 9709/71/M/J/11/3, Topic: -

Past experience has shown that the heights of a certain variety of rose bush have been normally distributed with mean $85.0 \mathrm{~cm}$. A new fertiliser is used and it is hoped that this will increase the heights. In order to test whether this is the case, a botanist records the heights, $x \mathrm{~cm}$, of a large random sample of $n$ rose bushes and calculates that $\bar{x}=85.7$ and $s=4.8$, where $\bar{x}$ is the sample mean and $s^{2}$ is an unbiased estimate of the population variance. The botanist then carries out an appropriate hypothesis test.

$\text{(i)}$ The test statistic, $z$, has a value of $1.786$ correct to 3 decimal places. Calculate the value of $n$. $[3]$

$\text{(ii)}$ Using this value of the test statistic, carry out the test at the $5 \%$ significance level. $[3]$

Question 2 Code: 9709/72/M/J/11/3, Topic: -

The number of goals scored per match by Everly Rovers is represented by the random variable $X$ which has mean $1.8$.

$\text{(i)}$ State two conditions for $X$ to be modelled by a Poisson distribution. $[2]$

Assume now that $X \sim \operatorname{Po}(1.8)$.

$\text{(ii)}$ Find $\mathrm{P}(2 < X < 6)$. $[2]$

$\text{(iii)}$ The manager promises the team a bonus if they score at least 1 goal in each of the next 10 matches. Find the probability that they win the bonus. $[3]$

Question 3 Code: 9709/73/M/J/11/3, Topic: -

At an election in $2010,15 \%$ of voters in Bratfield voted for the Renewal Party. One year later, a researcher asked 30 randomly selected voters in Bratfield whether they would vote for the Renewal Party if there were an election next week. 2 of these 30 voters said that they would.

$\text{(i)}$ Use a binomial distribution to test, at the $4 \%$ significance level, the null hypothesis that there has been no change in the support for the Renewal Party in Bratfield against the alternative hypothesis that there has been a decrease in support since the 2010 election. $[4]$

$\text{(ii)}~~$ $\text{(a)}$ Explain why the conclusion in part $\text{(i)}$ cannot involve a Type I error. $[1]$

$\text{(b)}$ State the circumstances in which the conclusion in part $\text{(i)}$ would involve a Type II error. $[1]$

Question 4 Code: 9709/71/O/N/11/3, Topic: -

Three coats of paint are sprayed onto a surface. The thicknesses, in millimetres, of the three coats have independent distributions $\mathrm{N}\left(0.13,0.02^{2}\right), \mathrm{N}\left(0.14,0.03^{2}\right)$ and $\mathrm{N}\left(0.10,0.01^{2}\right)$. Find the probability that, at a randomly chosen place on the surface, the total thickness of the three coats of paint is less than $0.30$ millimetres. $[5]$

Question 5 Code: 9709/72/O/N/11/3, Topic: -

Question 6 Code: 9709/73/O/N/11/3, Topic: -

Jack has to choose a random sample of 8 people from the 750 members of a sports club.

$\text{(i)}$ Explain fully how he can use random numbers to choose the sample. $[3]$

Jack asks each person in the sample how much they spent last week in the club café. The results, in dollars, were as follows.

$$ \begin{array}{llllllll} 15 & 25 & 30 & 8 & 12 & 18 & 27 & 25 \end{array} $$

$\text{(ii)}$ Find unbiased estimates of the population mean and variance. $[3]$

$\text{(iii)}$ Explain briefly what is meant by 'population' in this question. $[1]$

Worked solutions: P1, P3 & P6 (S1)

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