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Name of student | Date | ||||

Adm. number | Year/grade | Stream | |||

Subject | Probability & Statistics 2 (S2) | Variant(s) | P73 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | Total |
---|---|---|---|---|---|

Marks | 3 | 6 | 9 | 14 | 32 |

Score |

Question 1 Code: 9709/73/M/J/19/1, Topic: -

A coin is thrown 100 times and it shows heads 60 times. Calculate an approximate $98 \%$ confidence interval for the probability, $p$, that the coin shows heads on any throw. $[3]$

Question 2 Code: 9709/73/M/J/21/3, Topic: -

The local council claims that the average number of accidents per year on a particular road is $0.8$. Jane claims that the true average is greater than $0.8$. She looks at the records for a random sample of 3 recent years and finds that the total number of accidents during those 3 years was 5.

$\text{(a)}$ Assume that the number of accidents per year follows a Poisson distribution.

$\text{(i)}$ State null and alternative hypotheses for a test of Jane's claim. $[1]$

$\text{(ii)}$ Test at the $5 \%$ significance level whether Jane's claim is justified. $[4]$

$\text{(b)}$ Jane finds that the number of accidents per year has been gradually increasing over recent years.

State how this might affect the validity of the test carried out in part $\text{$\text{(a)}$ (ii)}$. $[1]$

Question 3 Code: 9709/73/M/J/10/6, Topic: -

Yu Ming travels to work and returns home once each day. The times, in minutes, that he takes to travel to work and to return home are represented by the independent random variables $W$ and $H$ with distributions $\mathrm{N}\left(22.4,4.8^{2}\right)$ and $\mathrm{N}\left(20.3,5.2^{2}\right)$ respectively.

$\text{(i)}$ Find the probability that Yu Ming's total travelling time during a 5-day period is greater than 180 minutes. $[4]$

$\text{(ii)}$ Find the probability that, on a particular day, Yu Ming takes longer to return home than he takes to travel to work. $[5]$

Question 4 Code: 9709/73/M/J/13/7, Topic: -

In the past the weekly profit at a store had mean $\$ 34600$ and standard deviation $\$ 4500$. Following a change of ownership, the mean weekly profit for 90 randomly chosen weeks was $\$ 35400$.

$\text{(i)}$ Stating a necessary assumption, test at the $5 \%$ significance level whether the mean weekly profit has increased. $[6]$

$\text{(ii)}$ State, with a reason, whether it was necessary to use the Central Limit theorem in part $\text{(i)}$. $[2]$

The mean weekly profit for another random sample of 90 weeks is found and the same test is carried out at the $5 \%$ significance level.

$\text{(iii)}$ State the probability of a Type I error. $[1]$

$\text{(iv)}$ Given that the population mean weekly profit is now $\$ 36500$, calculate the probability of a Type II error. $[5]$