AQA Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) 2022 June

1 The discrete random variable \(X\) has the following probability distribution Find \(\mathrm { P } ( X > 18 )\)
Circle your answer.
0.1
0.2
0.7
0.8

2 The continuous random variable \(Y\) has probability density function \(\mathrm { f } ( y )\) where $$\int _ { - \infty } ^ { \infty } y \mathrm { f } ( y ) \mathrm { d } y = 16 \text { and } \int _ { - \infty } ^ { \infty } y ^ { 2 } \mathrm { f } ( y ) \mathrm { d } y = 1040$$ Find the standard deviation of \(Y\) Circle your answer.
[0pt] [1 mark]
28
32
784
1024

3 The discrete random variable \(A\) has the following probability distribution function $$\mathrm { P } ( A = a ) = \begin{cases} 0.45 & a = 0
0.25 & a = 1
0.3 & a = 2
0 & \text { otherwise } \end{cases}$$ 3

1. Find the median of \(A\)
   3
2. Find the standard deviation of \(A\), giving your answer to three significant figures.
   3
3. \(\quad\) Find \(\operatorname { Var } ( 9 A - 2 )\)

4 The height of lilac trees, in metres, can be modelled by a normal distribution with variance 0.7 A random sample of \(n\) lilac trees is taken and used to construct a 99\% confidence interval for the population mean. This confidence interval is \(( 5.239,5.429 )\)
4

1. Find the value of \(n\)
   4
2. Joey claims that the mean height of lilac trees is 5.3 metres.
   State, with a reason, whether the confidence interval supports Joey's claim.

5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} x ^ { 3 } & 0 < x \leq 1
\frac { 9 } { 1696 } x ^ { 3 } \left( x ^ { 2 } + 1 \right) & 1 < x \leq 3
0 & \text { otherwise } \end{cases}$$ 5

1. Find \(\mathrm { P } ( X < 1.8 )\), giving your answer to three decimal places.
   [0pt] [3 marks]
   5
2. Find the lower quartile of \(X\)
   

   5	5 Show that \(\mathrm { E } \left( \frac { 1 } { X ^ { 2 } } \right) = \frac { 133 } { 212 }\)

6 The number of computers sold per day by a shop can be modelled by the random variable \(Y\) where \(Y \sim \operatorname { Po } ( 42 )\) 6

1. State the variance of \(Y\) 6
2. One month ago, the shop started selling a new model of computer.
   On a randomly chosen day in the last month, the shop sold 53 computers.
   Carry out a hypothesis test, at the \(5 \%\) level of significance, to investigate whether the mean number of computers sold per day has increased in the last month.
   [0pt] [6 marks]
   6
3. Describe, in the context of the hypothesis test in part (b), what is meant by a Type II error.

7 Wade and Odelia are investigating whether there is an association between the region where a person lives and the brand of washing powder they use. They decide to conduct a \(\chi ^ { 2 }\)-test for association and survey a random sample of 200 people. The expected frequencies for the test have been calculated and are shown in the contingency table below.