AQA Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) 2019 June

1 The discrete random variable \(X\) has the following probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} \frac { 5 - x } { 10 } & x = 1,2,3,4
0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X \geq 3 )\)
Circle your answer.
0.1
0.15
0.2
0.3

2 A binomial hypothesis test was carried out at the \(5 \%\) level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : p = 0.6
& \mathrm { H } _ { 1 } : p > 0.6 \end{aligned}$$ A sample of size 30 was used to carry out the test.
Find the probability that a Type I error was made.
Circle your answer.
[0pt] [1 mark]
\(4.4 \%\) 4.8\% 5.0\% 9.4\%

3 Fiona is studying the heights of corn plants on a farm. She measures the height, \(x \mathrm {~cm}\), of a random sample of 200 corn plants on the farm.
The summarised results are as follows: $$\sum x = 60255 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 995$$ Calculate a \(96 \%\) confidence interval for the population mean of heights of corn plants on the farm, giving your values to one decimal place.
\begin{center} \begin{tabular}{|l|l|l|l|} \hline \multicolumn{3}{|l|}{\begin{tabular}{l} \(\begin{aligned} & 4 \text { The continuous random variable } X \text { has probability density fu }
& \qquad f ( x ) = \begin{cases} \frac { 4 } { 99 } \left( 12 x - x ^ { 2 } - x ^ { 3 } \right) & 0 \leq x \leq 3
0 & \text { otherwise } \end{cases} \end{aligned}\)

4

1. \(\text { Find } \mathrm { P } ( X > 1 )\)
   [0pt] [3 marks]
   4
2. 
   [0pt] [3 marks]
   \end{tabular}} & Do not write outside the box
   \hline \end{tabular} \end{center} □
   4
3. Find \(\mathrm { E } \left( 2 X ^ { - 1 } - 3 \right)\)

5 The discrete random variable \(X\) has the following probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} \frac { 1 } { n } & x = 1,2 , \ldots , n
0 & \text { otherwise } \end{cases}$$ 5

1. 1. Prove that \(\mathrm { E } ( X ) = \frac { n + 1 } { 2 }\)
      [0pt] [3 marks]
      5
2. (ii) Prove that \(\operatorname { Var } ( X ) = \frac { n ^ { 2 } - 1 } { 12 }\)
   

   5	State two conditions under which a discrete uniform distribution can be used to model the score when a cubic dice is rolled. [2 marks]

6 A company owns two machines, \(A\) and \(B\), which make toys. Both machines run continuously and independently. Machine \(A\) makes an average of 2 errors per hour.
6

1. Using a Poisson model, find the probability that the machine makes exactly 5 errors in 4 hours, giving your answer to three significant figures. 6
2. Machine \(B\) makes an average of 5 errors per hour. Both machines are switched on and run for 1 hour. The company finds the probability that the total number of errors made by machines \(A\) and \(B\) in 1 hour is greater than 8 . If the probability is greater than 0.4 , a new machine will be purchased.
   Using a Poisson model, determine whether or not the toy company will purchase a new machine.
   6
3. After investigation, the standard deviation of errors made by machine \(A\) is found to be 0.5 errors per hour and the standard deviation of errors made by machine \(B\) is also found to be 0.5 errors per hour. Explain whether or not the use of Poisson models in parts (a) and (b) is appropriate.

7 Mohammed is conducting a medical trial to study the effect of two drugs, \(A\) and \(B\), on the amount of time it takes to recover from a particular illness. Drug \(A\) is used by one group of 60 patients and drug \(B\) is used by a second group of 60 patients. The results are summarised in the table: