AQA Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) 2018 June

1 Let \(X\) be a continuous random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 } x ( 2 - x ) & 0 \leq x \leq 2
0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X = 1 )\)
Circle your answer.
0
\(\frac { 1 } { 2 }\)
\(\frac { 3 } { 4 }\)
\(\frac { 27 } { 32 }\)

2 The discrete random variable \(Y\) has a Poisson distribution with mean 3 Find the value of \(\mathrm { P } ( Y > 1 )\) to three significant figures.
Circle your answer.
\(0.149 \quad 0.199 \quad 0.801 \quad 0.950\)

3 The discrete random variable \(X\) has the following probability distribution The continuous random variable \(Y\) has the following probability density function $$\mathrm { f } ( y ) = \begin{cases} \frac { 1 } { 64 } y ^ { 3 } & 0 \leq y \leq 4
0 & \text { otherwise } \end{cases}$$ Given that \(X\) and \(Y\) are independent, show that \(\mathrm { E } \left( X ^ { 2 } + Y ^ { 2 } \right) = \frac { 1327 } { 60 }\)

4 The waiting times for patients to see a doctor in a hospital can be modelled with a normal distribution with known variance of 10 minutes. 4

1. A random sample of 100 patients has a total waiting time of 3540 minutes.
   Calculate a \(98 \%\) confidence interval for the population mean of waiting times, giving values to four significant figures.
   4
2. Dante conducts a hypothesis test with the sample from part (a) on the waiting times. Dante's hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 38
   & \mathrm { H } _ { 1 } : \mu \neq 38 \end{aligned}$$ Dante uses a \(2 \%\) level of significance.
   Explain whether Dante accepts or rejects the null hypothesis.

5 The diagram shows a graph of the probability density function of the random variable \(X\).
\includegraphics[max width=\textwidth, alt={}, center]{313cd5ce-07ff-4781-a134-565b8b221145-05_574_1086_406_479} 5

1. State the mode of \(X\).
   5
2. Find the probability density function of \(X\).

6 The discrete random variable \(Y\) has the probability function $$\mathrm { P } ( Y = y ) = \begin{cases} 2 k y & y = 1,2,3,4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. Show that \(\operatorname { Var } ( 5 Y - 2 ) = 25\)
\includegraphics[max width=\textwidth, alt={}, center]{313cd5ce-07ff-4781-a134-565b8b221145-07_2488_1716_219_153}

7 Over a period of time it has been shown that the mean number of vehicles passing a service station on a motorway is 50 per minute. After a new motorway junction was built nearby, Xander observed that 30 vehicles passed the service station in one minute. 7

1. Xander claims that the construction of the new motorway junction has reduced the mean number of vehicles passing the service station per minute. Investigate Xander's claim, using a suitable test at the \(1 \%\) level of significance.
   7
2. For your test carried out in part (a) state, in context, the meaning of a Type 1 error. 7
3. Explain why the model used in part (a) might be invalid.

8 An insurance company groups its vehicle insurance policies into two categories, car insurance and motorbike insurance. The number of claims in a random sample of 80 policies was monitored and the results summarised in contingency Table 1. \begin{table}[h] \end{table} The insurance company decides to carry out a \(\chi ^ { 2 }\)-test for association between number of claims and type of insurance policy using the information given in Table 1. 8

1. The contingency table shown in Table 2 gives some of the exact expected frequencies for this test. Complete Table 2 with the missing exact expected values. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 2}

   \multirow{2}{*}{}		Number of claims
   		0	1	2	3 or more
   \multirow{2}{*}{Type of insurance policy}	Car		10.0625		4.375
   	Motorbike			10.6875

   \end{table} 8
2. Carry out the insurance company's test, using the \(10 \%\) level of significance.
   \includegraphics[max width=\textwidth, alt={}, center]{313cd5ce-07ff-4781-a134-565b8b221145-12_2488_1719_219_150} Additional page, if required.
   Write the question numbers in the left-hand margin. Additional page, if required.
   Write the question numbers in the left-hand margin. Additional page, if required.
   Write the question numbers in the left-hand margin.