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### MATHEMATICS 9709

#### Cambridge International AS and A Level

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

Qtn No. 1 2 3 4 Total
Marks 4 5 9 11 29
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 4 questions Question 1 Code: 9709/71/M/J/11/1, Topic: - On average, 2 people in every 10000 in the UK have a particular gene. A random sample of 6000 people in the UK is chosen. The random variable$X$denotes the number of people in the sample who have the gene. Use an approximating distribution to calculate the probability that there will be more than 2 people in the sample who have the gene.$[4]$Question 2 Code: 9709/71/M/J/11/2, Topic: -$\text{(a)}$The time taken by a worker to complete a task was recorded for a random sample of 50 workers. The sample mean was$41.2$minutes and an unbiased estimate of the population variance was$32.6$minutes$^{2}$. Find a$95 \%$confidence interval for the mean time taken to complete the task.$[3]\text{(b)}$The probability that an$\alpha \%$confidence interval includes only values that are lower than the population mean is$\frac{1}{16}$. Find the value of$\alpha$.$[2]$Question 3 Code: 9709/71/M/J/18/5, Topic: - The mass, in kilograms, of rocks in a certain area has mean$14.2$and standard deviation$3.1.\text{(i)}$Find the probability that the mean mass of a random sample of 50 of these rocks is less than$14.0 \mathrm{~kg}$.$[3]\text{(ii)}$Explain whether it was necessary to assume that the population of the masses of these rocks is normally distributed.$[1]\text{(iii)}$A geologist suspects that rocks in another area have a mean mass which is less than$14.2 \mathrm{~kg}$. A random sample of 100 rocks in this area has sample mean$13.5 \mathrm{~kg}$. Assuming that the standard deviation for rocks in this area is also$3.1 \mathrm{~kg}$, test at the$2 \%$significance level whether the geologist is correct.$[5]$Question 4 Code: 9709/71/M/J/18/6, Topic: - The time, in minutes, taken by people to complete a test is modelled by the continuous random variable$X$with probability density function given by $$\mathrm{f}(x)= \begin{cases}\dfrac{k}{x^{2}} & 5 \leqslant x \leqslant 10 \\ 0 & \text { otherwise }\end{cases}$$ where$k$is a constant.$\text{(i)}$Show that$k=10$.$[3]\text{(ii)}$Show that$\mathrm{E}(X)=10 \ln 2$.$[2]\text{(iii)}$Find$\mathrm{P}(X>9)$.$[3]\text{(iv)}$Given that$\mathrm{P}(X < a)=0.6$, find$a$.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

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