AQA Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) 2024 June

1 The discrete random variable \(X\) has probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} 0.45 & x = 1
0.25 & x = 2
0.25 & x = 3
0.05 & x = 4
0 & \text { otherwise } \end{cases}$$ State the mode of \(X\) Circle your answer.
0.25
0.45
1
2.5

2 A test for association is to be carried out. The tables below show the observed frequencies and the expected frequencies that are to be used for the test. It is necessary to merge some rows or columns before the test can be carried out.
Find the entry in the tables that provides evidence for this.
Circle your answer.
[0pt] [1 mark]
Observed A-Z
Observed B-Z
Expected A-X
Expected B-Y

3 The random variable \(X\) has a normal distribution with known variance 15.7 A random sample of size 120 is taken from \(X\) The sample mean is 68.2 Find a 94\% confidence interval for the population mean of \(X\) Give your limits to three significant figures.

4 The discrete random variable \(Y\) has probability distribution The standard deviation of \(Y\) is \(s\) 4

1. Show that \(s = 10.53\) correct to two decimal places.
   [0pt] [4 marks]
   4
2. The median of \(Y\) is \(m\) Find \(\mathrm { P } ( Y > m - 1.5 s )\)

5 A spinner has 8 equal areas numbered 1 to 8, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{de9f0107-38de-4d0d-8391-4d29b98fa601-06_383_390_319_810} The spinner is spun and lands with one of its edges on the ground. 5

1. Assume that the spinner lands on each number with equal probability. 5
2. 1. State a distribution that could be used to model the number that the spinner lands on. 5
3. (ii) Use your distribution from part 5
4. 1. to find the probability that the spinner lands on a number greater than 5
      [0pt] [1 mark] 5
5. Clare spins the spinner 1000 times and records the results in the following table.

   Number landed on	1	2	3	4	5	6	7	8
   Frequency	37	64	112	161	308	156	109	53

   5
6. 1. Explain how the data shows that the model used in part (a) may not be valid.
      5
7. (ii) Describe how Clare's results could be used to adjust the model.

6 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 x } { 44 } + \frac { 1 } { 22 } & 1 \leq x \leq 5
0 & \text { otherwise } \end{cases}$$ 6

1. Find \(\mathrm { P } ( X > 2 )\)
   [0pt] [2 marks]
   6
2. Find the upper quartile of \(X\) Give your answer to two decimal places.
   6
3. Find \(\operatorname { Var } \left( 44 X ^ { - 3 } \right)\) Give your answer to three decimal places.

7 Over a period of time, it has been shown that the mean number of customers entering a small store is 6 per hour. The store runs a promotion, selling many products at lower prices. 7

1. Luke randomly selects an hour during the promotion and counts 11 customers entering the store. He claims that the promotion has changed the mean number of customers per hour entering the store. Investigate Luke's claim, using the \(5 \%\) level of significance.
   7
2. Luke randomly selects another hour and carries out the same investigation as in part (a). Find the probability of a Type I error, giving your answer to four decimal places.
   Fully justify your answer.
   7
3. When observing the store, Luke notices that some customers enter the store together as a group. Explain why the model used in parts (a) and (b) might not be valid.
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