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Name of student | Date | ||||

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

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

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 5 | 3 | 3 | 4 | 4 | 6 | 4 | 8 | 10 | 13 | 9 | 12 | 81 |

Score |

Question 1 Code: 9709/71/O/N/14/1, Topic: -

The masses, in grams, of potatoes of types $A$ and $B$ have the distributions $N\left(175,60^{2}\right)$ and $N\left(105,28^{2}\right)$ respectively. Find the probability that a randomly chosen potato of type $A$ has a mass that is at least twice the mass of a randomly chosen potato of type $B$. $[5]$

Question 2 Code: 9709/71/O/N/16/1, Topic: -

The weights, in kilograms, of a random sample of eight 16 -year old males are given below.

$$ \begin{array}{llllllll} 58.9 & 63.5 & 62.7 & 59.4 & 66.9 & 68.0 & 60.4 & 68.2 \end{array} $$Find unbiased estimates of the population mean and variance of the weights of all 16 -year old males. $[3]$

Question 3 Code: 9709/73/O/N/16/1, Topic: -

The random variable $X$ has the distribution $\operatorname{Po}(3.5)$. Find $\mathrm{P}(X < 3)$. $[3]$

Question 4 Code: 9709/73/O/N/18/1, Topic: -

71Question 5 Code: 9709/71/O/N/17/2, Topic: -

An airline has found that, on average, 1 in 100 passengers do not arrive for each flight, and that this occurs randomly. For one particular flight the airline always sells 403 seats. The plane only has room for 400 passengers, so the flight is overbooked if the number of passengers who do not arrive is less than 3. Use a suitable approximation to find the probability that the flight is overbooked. $[4]$

Question 6 Code: 9709/71/O/N/15/3, Topic: -

Jagdeesh measured the lengths, $x$ minutes, of 60 randomly chosen lectures. His results are summarised below.

$$ n=60 \quad \Sigma x=3420 \quad \Sigma x^{2}=195200 $$$\text{(i)}$ Calculate unbiased estimates of the population mean and variance. $[3]$

$\text{(ii)}$ Calculate a $98 \%$ confidence interval for the population mean. $[3]$

Question 7 Code: 9709/73/O/N/17/3, Topic: -

Question 8 Code: 9709/71/O/N/12/4, Topic: -

A cereal manufacturer claims that $25 \%$ of cereal packets contain a free gift. Lola suspects that the true proportion is less than $25 \%$. In order to test the manufacturer's claim at the $5 \%$ significance level, she checks a random sample of 20 packets.

$\text{(i)}$ Find the critical region for the test. $[5]$

$\text{(ii)}$ Hence find the probability of a Type I error. $[1]$

Lola finds that 2 packets in her sample contain a free gift.

$\text{(iii)}$ State, with a reason, the conclusion she should draw. $[2]$

Question 9 Code: 9709/71/O/N/14/4, Topic: -

In a survey a random sample of 150 households in Nantville were asked to fill in a questionnaire about household budgeting.

$\text{(i)}$ The results showed that 33 households owned more than one car. Find an approximate $99 \%$ confidence interval for the proportion of all households in Nantville with more than one car. $[4]$

$\text{(ii)}$ The results also included the weekly expenditure on food, $x$ dollars, of the households. These were summarised as follows.

$$ n=150 \quad \Sigma x=19035 \quad \Sigma x^{2}=4054716 $$Find unbiased estimates of the mean and variance of the weekly expenditure on food of all households in Nantville. $[3]$

$\text{(iii)}$ The government has a list of all the households in Nantville numbered from 1 to 9526. Describe briefly how to use random numbers to select a sample of 150 households from this list. $[3]$

Question 10 Code: 9709/73/O/N/20/5, Topic: -

Question 11 Code: 9709/73/O/N/15/6, Topic: -

Parcels arriving at a certain office have weights $W \mathrm{~kg}$, where the random variable $W$ has mean $\mu$ and standard deviation $0.2$. The value of $\mu$ used to be $2.60$, but there is a suspicion that this may no longer be true. In order to test at the $5 \%$ significance level whether the value of $\mu$ has increased, a random sample of 75 parcels is chosen. You may assume that the standard deviation of $W$ is unchanged.$\text{(i)}$ The mean weight of the 75 parcels is found to be $2.64 \mathrm{~kg}$. Carry out the test. $[4]$

$\text{(ii)}$ Later another test of the same hypotheses at the $5 \%$ significance level, with another random sample of 75 parcels, is carried out. Given that the value of $\mu$ is now $2.68$, calculate the probability of a Type II error. $[5]$

Question 12 Code: 9709/73/O/N/20/6, Topic: -