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Name of student | HENRYTAIGO | Date | |||

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

Subject | Probability & Statistics 2 (S2) | Variant(s) | P71, P72, P73 | ||

Start time | Duration | Stop time |

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

Marks | 11 | 10 | 10 | 31 |

Score |

Question 1 Code: 9709/72/M/J/19/7, Topic: -

All the seats on a certain daily flight are always sold. The number of passengers who have bought seats but fail to arrive for this flight on a particular day is modelled by the distribution $\mathrm{B}(320,0.005)$.

$\text{(i)}$ Explain what the number 320 represents in this context. $[1]$

$\text{(ii)}$ The total number of passengers who have bought seats but fail to arrive for this flight on 2 randomly chosen days is denoted by $X$. Use a suitable approximating distribution to find $\mathrm{P}(2 < X < 6)$ $[3]$

$\text{(iii)}$ Justify the use of your approximating distribution. $[2]$

After some changes, the airline wishes to test whether the mean number of passengers per day who fail to arrive for this flight has decreased.

$\text{(iv)}$ During 5 randomly chosen days, a total of 2 passengers failed to arrive. Carry out the test at the $2.5 \%$ significance level. $[5]$

Question 2 Code: 9709/73/M/J/19/7, Topic: -

Each day at a certain doctor's surgery there are 70 appointments available in the morning and 60 in the afternoon. All the appointments are filled every day. The probability that any patient misses a particular morning appointment is $0.04$, and the probability that any patient misses a particular afternoon appointment is $0.05$. All missed appointments are independent of each other.

Use suitable approximating distributions to answer the following.

$\text{(i)}$ Find the probability that on a randomly chosen morning there are at least 3 missed appointments. $[3]$

$\text{(ii)}$ Find the probability that on a randomly chosen day there are a total of exactly 6 missed appointments. $[3]$

$\text{(iii)}$ Find the probability that in a randomly chosen 10-day period there are more than $50 \mathrm{missed}$ appointments. $[3]$

Question 3 Code: 9709/73/O/N/19/7, Topic: -

Bob is a self-employed builder. In the past his weekly income had mean $\$ 546$ and standard deviation $\$ 120$. Following a change in Bob's working pattern, his mean weekly income for 40 randomly chosen weeks was $\$ 581$. You should assume that the standard deviation remains unchanged at $\$ 120$.

$\text{(i)}$ Test at the $2.5 \%$ significance level whether Bob's mean weekly income has increased. $[5]$

Bob finds his mean weekly income for another random sample of 40 weeks and carries out a similar test at the $2.5 \%$ significance level.

$\text{(ii)}$ Given that Bob's mean weekly income is now in fact $\$ 595$, find the probability of a Type II error. $[5]$