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Qtn No.	1	2	3	4	5	Total
Marks	8	8	11	9	9	45
Score

Question 1 Code: 9709/61/M/J/11/5, Topic: The normal distribution

$\text{(a)}$ The random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$. It is given that $3 \mu=7 \sigma^{2}$ and that $\mathrm{P}(X>2 \mu)=0.1016$. Find $\mu$ and $\sigma$. $[4]$

$\text{(b)}$ It is given that $Y \sim \mathrm{N}(33,21)$. Find the value of $a$ given that $\mathrm{P}(33-a$[4]$

Question 2 Code: 9709/62/M/J/11/5, Topic: Representation of data

A hotel has 90 rooms. The table summarises information about the number of rooms occupied each day for a period of 200 days.

$$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text{Number of rooms occupied} & 1-20 & 21-40 & 41-50 & 51-60 & 61-70 & 71-90 \\ \hline \text{Frequency} & 10 & 32 & 62 & 50 & 28 & 18 \\ \hline \end{array} $$

$\text{(i)}$ Draw a cumulative frequency graph on graph paper to illustrate this information. $[4]$

$\text{(ii)}$ Estimate the number of days when over 30 rooms were occupied. $[2]$

$\text{(iii)}$ On $75 \%$ of the days at most $n$ rooms were occupied. Estimate the value of $n$. $[2]$

Question 3 Code: 9709/63/M/J/11/5, Topic: The normal distribution

The random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\frac{1}{4} \mu$. It is given that $\mathrm{P}(X>20)=0.04$.

$\text{(i)}$ Find $\mu$. $[3]$

$\text{(ii)}$ Find $\mathrm{P}(10 < X < 20)$. $[3]$

$\text{(iii)}$ 250 independent observations of $X$ are taken. Find the probability that at least 235 of them are less than 20. $[5]$

Question 4 Code: 9709/61/O/N/11/5, Topic: The normal distribution

The weights of letters posted by a certain business are normally distributed with mean $20 \mathrm{~g}$. It is found that the weights of $94 \%$ of the letters are within $12 \mathrm{~g}$ of the mean.

$\text{(i)}$ Find the standard deviation of the weights of the letters. $[3]$

$\text{(ii)}$ Find the probability that a randomly chosen letter weighs more than $13 \mathrm{~g}$. $[3]$

$\text{(iii)}$ Find the probability that at least 2 of a random sample of 7 letters have weights which are more than $12 \mathrm{~g}$ above the mean. $[3]$

Question 5 Code: 9709/62/O/N/11/5, Topic: Discrete random variables

A triangular spinner has one red side, one blue side and one green side. The red side is weighted so that the spinner is four times more likely to land on the red side than on the blue side. The green side is weighted so that the spinner is three times more likely to land on the green side than on the blue side.

$\text{(i)}$ Show that the probability that the spinner lands on the blue side is $\frac{1}{8}$. $[1]$

$\text{(ii)}$ The spinner is spun 3 times. Find the probability that it lands on a different coloured side each time. $[3]$

$\text{(iii)}$ The spinner is spun 136 times. Use a suitable approximation to find the probability that it lands on the blue side fewer than 20 times. $[5]$