AQA Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) 2021 June

1 The discrete random variable \(X\) has \(\operatorname { Var } ( X ) = 6.5\)
Find \(\operatorname { Var } ( 4 X - 2 )\) Circle your answer.
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2 The random variable \(A\) has a Poisson distribution with mean 2 The random variable \(B\) has a Poisson distribution with standard deviation 4 The random variables \(A\) and \(B\) are independent.
State the distribution of \(A + B\) Circle your answer.
[0pt] [1 mark]
Po(4)
Po(6)
Po(8)
Po(18)

3 The random variable \(X\) has a discrete uniform distribution and takes values \(1,2,3 , \ldots , n\) The mean of \(X\) is 8 3

1. Show that \(n = 15\)
   [0pt] [2 marks]
   LL
   3
2. \(\quad\) Find \(\mathrm { P } ( X > 4 )\)
   3
3. Find the variance of \(X\), giving your answer in exact form.

4 The distance a particular football player runs in a match is modelled by a normal distribution with standard deviation 0.3 kilometres. A random sample of \(n\) matches is taken.
The distance the player runs in this sample of matches has mean 10.8 kilometres.
The sample is used to construct a \(93 \%\) confidence interval for the mean, of width 0.0543 kilometres, correct to four decimal places. 4

1. Find the value of \(n\)
   4
2. Find the \(93 \%\) confidence interval for the mean, giving the limits to three decimal places.
   4
3. Alison claims that the population mean distance the player runs is 10.7 kilometres. She carries out a hypothesis test at the 7\% level of significance using the random sample and the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 10.7
   & \mathrm { H } _ { 1 } : \mu \neq 10.7 \end{aligned}$$ 4
4. 1. State, with a reason, whether the null hypothesis will be accepted or rejected. 4
5. (ii) Describe, in the context of the hypothesis test in part (c)(i), what is meant by a Type II error.
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5 In a game it is known that:

• 25\% of players score 0
• 30\% of players score 5
• 35\% of players score 10
• 10\% of players score 20

Players receive prize money, in pounds, equal to 100 times their score.
5

1. State the modal score.
   [0pt] [1 mark] 5
2. Find the median score.
   5
3. Find the mean prize money received by a player.

6 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 114 } ( 4 x + 7 ) & 0 \leq x \leq 6
0 & \text { otherwise } \end{cases}$$ 6

1. Show that the median of \(X\) is 3.87, correct to three significant figures.
   [0pt] [3 marks]
   6
2. Find the exact value of \(\mathrm { P } ( X > 2 )\)
   

   	6 The continuous random variable \(Y\) has probability density function \(g ( y ) = \begin{cases} \frac { 1 } { 2 } y ^ { 2 } - \frac { 1 } { 6 } y ^ { 3 } 1 \leq y \leq 3 0 \text { otherwise } \end{cases}\) " 6 Show that \(\operatorname { Var } \left( \frac { 1 } { Y } \right) = \frac { 2 } { 81 }\)	\multirow[b]{2}{*}{ [4 marks] [4 marks] }

7 Two employees, \(A\) and \(B\), both produce the same toy for a company. The company records the total number of errors made per day by each employee during a 40-day period. The results are summarised in the following table. Employee The company claims that there is an association between employee and number of errors made per day. 7

1. Test the company's claim, using the \(5 \%\) level of significance.
   7
2. By considering observed and expected frequencies, interpret in context the association between employee and number of errors made per day.
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