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Name of student | Date | ||||

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

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

Start time | Duration | Stop time |

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

Marks | 2 | 4 | 5 | 4 | 10 | 8 | 6 | 10 | 10 | 11 | 6 | 10 | 86 |

Score |

Question 1 Code: 9709/71/M/J/10/1, Topic: -

Fred arrives at random times on a station platform. The times in minutes he has to wait for the next train are modelled by the continuous random variable for which the probability density function $\mathrm{f}$ is shown above.

$\text{(i)}$ State the value of $k$. $[1]$

$\text{(ii)}$ Explain briefly what this graph tells you about the arrival times of trains. $[1]$

Question 2 Code: 9709/71/M/J/14/1, Topic: -

The masses, in grams, of apples of a certain type are normally distributed with mean $60.4$ and standard deviation 8.2. The apples are packed in bags, with each bag containing 8 randomly chosen apples. The bags are checked by Quality Control and any bag containing apples with a total mass of less than $436 \mathrm{~g}$ is rejected. Find the proportion of bags that are rejected. $[4]$

Question 3 Code: 9709/71/M/J/19/1, Topic: -

At an internet cafe, the charge for using a computer is 5 cents per minute. The number of minutes for which people use a computer has mean 23 and standard deviation $8.$

$\text{(i)}$ Find, in cents, the mean and standard deviation of the amount people pay when using a computer. $[2]$

$\text{(ii)}$ Each day, 15 people use computers independently. Find, in cents, the mean and standard deviation of the total amount paid by 15 people. $[3]$

Question 4 Code: 9709/71/M/J/21/2, Topic: -

The time, in minutes, taken by students to complete a test has the distribution $\mathrm{N}(125,36)$.

$\text{(a)}$ Find the probability that the mean time taken to complete the test by a random sample of 40 students is less than 123 minutes. $[3]$

$\text{(b)}$ Explain whether it was necessary to use the Central Limit theorem in the solution to part $\text{(a)}$. $[1]$

Question 5 Code: 9709/71/M/J/11/4, Topic: -

$\text{(a)}$

The diagrams show the graphs of two functions, $\mathrm{g}$ and $\mathrm{h}$. For each of the functions $\mathrm{g}$ and $\mathrm{h}$, give a reason why it cannot be a probability density function. $[3]$

$\text{(b)}$ The distance, in kilometres, travelled in a given time by a cyclist is represented by the continuous random variable $X$ with probability density function given by

$$ \mathrm{f}(x)= \begin{cases}\dfrac{30}{x^{2}} & 10 \leqslant x \leqslant 15 \\ 0 & \text { otherwise }\end{cases} $$$\text{(i)}$ Show that $\mathrm{E}(X)=30 \ln 1.5$. $[3]$

$\text{(ii)}$ Find the median of $X$. Find also the probability that $X$ lies between the median and the mean. $[5]$

Question 6 Code: 9709/71/M/J/13/4, Topic: -

The lengths, $x \mathrm{~m}$, of a random sample of 200 balls of string are found and the results are summarised by $\Sigma x=2005$ and $\Sigma x^{2}=20175$.

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

$\text{(ii)}$ Use the values from part $\text{(i)}$ to estimate the probability that the mean length of a random sample of 50 balls of string is less than $10 \mathrm{~m}$. $[3]$

$\text{(iii)}$ Explain whether or not it was necessary to use the Central Limit theorem in your calculation in part $\text{(ii)}$. $[2]$

Question 7 Code: 9709/71/M/J/20/4, Topic: -

A fair spinner has five sides numbered $1,2,3,4,5$. The score on one spin is denoted by $X$.

$\text{(a)}$ Show that $\operatorname{Var}(X)=2$. $[1]$

Fiona has another spinner, also with five sides numbered $1,2,3,4,5$. She suspects that it is biased so that the expected score is less than 3. In order to test her suspicion, she plans to spin her spinner 40 times. If the mean score is less than $2.6$ she will conclude that her spinner is biased in this way.

$\text{(b)}$ Find the probability of a Type I error. $[4]$

$\text{(c)}$ State what is meant by a Type II error in this context. $[1]$

Question 8 Code: 9709/71/M/J/12/5, Topic: -

A random variable $X$ has the distribution $\operatorname{Po}(3.2)$.

$\text{(i)}$ A random value of $X$ is found.

$\text{(a)}$ Find $\mathrm{P}(X \geqslant 3)$. $[2]$

$\text{(b)}$ Find the probability that $X=3$ given that $X \geqslant 3$. $[3]$

$\text{(ii)}$ Random samples of 120 values of $X$ are taken.

$\text{(a)}$ Describe fully the distribution of the sample mean. $[2]$

$\text{(b)}$ Find the probability that the mean of a random sample of size 120 is less than $3.3$. $[3]$

Question 9 Code: 9709/71/M/J/17/5, Topic: -

Large packets of sugar are packed in cartons, each containing 12 packets. The weights of these packets are normally distributed with mean $505 \mathrm{~g}$ and standard deviation $3.2 \mathrm{~g}$. The weights of the cartons, when empty, are independently normally distributed with mean $150 \mathrm{~g}$ and standard deviation $7 \mathrm{~g}$.

$\text{(i)}$ Find the probability that the total weight of a full carton is less than $6200 \mathrm{~g}$. $[5]$

Small packets of sugar are packed in boxes. The total weight of a full box has a normal distribution with mean $3130 \mathrm{~g}$ and standard deviation $12.1 \mathrm{~g}$.$\text{(ii)}$ Find the probability that the weight of a randomly chosen full carton is less than double the weight of a randomly chosen full box. $[5]$

Question 10 Code: 9709/71/M/J/12/6, Topic: -

A survey taken last year showed that the mean number of computers per household in Branley was $1.66$. This year a random sample of 50 households in Branley answered a questionnaire with the following results.

$$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text{Number of computers} & 0 & 1 & 2 & 3 & 4 & >4 \\ \hline \text{Number of households} & 5 & 12 & 18 & 10 & 5 & 0 \\ \hline \end{array} $$$\text{(i)}$ Calculate unbiased estimates for the population mean and variance of the number of computers per household in Branley this year. $[3]$

$\text{(ii)}$ Test at the $5 \%$ significance level whether the mean number of computers per household has changed since last year. $[5]$

$\text{(iii)}$ Explain whether it is possible that a Type I error may have been made in the test in part $\text{(ii)}$. $[1]$

$\text{(iv)}$ State what is meant by a Type II error in the context of the test in part $\text{(ii)}$, and give the set of values of the test statistic that could lead to a Type II error being made. $[2]$

Question 11 Code: 9709/71/M/J/21/6, Topic: -

The probability density function, f, of a random variable $X$ is given by

$$ \mathrm{f}(x)= \begin{cases}k\left(6 x-x^{2}\right) & 0 \leqslant x \leqslant 6 \\ 0 & \text { otherwise }\end{cases} $$where $k$ is a constant.

State the value of $\mathrm{E}(X)$ and show that $\operatorname{Var}(X)=\frac{9}{5}$. $[6]$

Question 12 Code: 9709/71/M/J/13/7, Topic: -

Leila suspects that a particular six-sided die is biased so that the probability, $p$, that it will show a six is greater than $\frac{1}{6}$. She tests the die by throwing it 5 times. If it shows a six on 3 or more throws she will conclude that it is biased.

$\text{(i)}$ State what is meant by a Type I error in this situation and calculate the probability of a Type I error. $[3]$

$\text{(ii)}$ Assuming that the value of $p$ is actually $\frac{2}{3}$, calculate the probability of a Type II error. $[3]$

Leila now throws the die 80 times and it shows a six on 50 throws.

$\text{(iii)}$ Calculate an approximate $96 \%$ confidence interval for $p$. $[4]$