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Qtn No.	1	2	3	4	Total
Marks	5	6	8	8	27
Score

Question 1 Code: 9709/71/M/J/13/3, Topic: -

Weights of cups have a normal distribution with mean $91 \mathrm{~g}$ and standard deviation $3.2 \mathrm{~g}$. Weights of saucers have an independent normal distribution with mean $72 \mathrm{~g}$ and standard deviation $2.6 \mathrm{~g}$. Cups and saucers are chosen at random to be packed in boxes, with 6 cups and 6 saucers in each box. Given that each empty box weighs $550 \mathrm{~g}$, find the probability that the total weight of a box containing 6 cups and 6 saucers exceeds $1550 \mathrm{~g}$. $[5]$

Question 2 Code: 9709/73/M/J/13/3, Topic: -

Each of a random sample of 15 students was asked how long they spent revising for an exam. The results, in minutes, were as follows.

$$ \begin{array}{lllllllllllllll} 50 & 70 & 80 & 60 & 65 & 110 & 10 & 70 & 75 & 60 & 65 & 45 & 50 & 70 & 50 \end{array} $$

Assume that the times for all students are normally distributed with mean $\mu$ minutes and standard deviation 12 minutes.

$\text{(i)}$ Calculate a $92 \%$ confidence interval for $\mu$. $[4]$

$\text{(ii)}$ Explain what is meant by a $92 \%$ confidence interval for $\mu$. $[1]$

$\text{(iii)}$ Explain what is meant by saying that a sample is 'random'. $[1]$

Question 3 Code: 9709/71/O/N/13/3, Topic: -

Following a change in flight schedules, an airline pilot wished to test whether the mean distance that he flies in a week has changed. He noted the distances, $x \mathrm{~km}$, that he flew in 50 randomly chosen weeks and summarised the results as follows.

$$ n=50 \quad \Sigma x=143~300 \quad \Sigma x^{2}=410~900~000 $$

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

$\text{(ii)}$ In the past, the mean distance that he flew in a week was $2850 \mathrm{~km}$. Test, at the $5 \%$ significance level, whether the mean distance has changed. $[5]$

Question 4 Code: 9709/72/O/N/13/3, Topic: -