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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 |

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 $a$. $[3]$