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### HENRYTAIGO

#### 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
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions 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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