Questions — AQA Further AS Paper 2 Statistics (59 questions)

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AQA Further AS Paper 2 Statistics 2023 June Q7
7 A theatre has morning, afternoon and evening shows. On one particular day, the theatre asks all of its customers to state whether they enjoyed or did not enjoy the show. The results are summarised in the table.
Morning showAfternoon showEvening showTotal
Enjoyed6291172325
Not enjoyed2535115175
Total87126287500
The theatre claims that there is no association between the show that a customer attends and whether they enjoyed the show. 7
  1. Investigate the theatre's claim, using a \(2.5 \%\) level of significance.
    7
  2. By considering observed and expected frequencies, interpret in context the association between the show that a customer attends and whether they enjoyed the show.
AQA Further AS Paper 2 Statistics 2023 June Q8
8 The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) It is given that \(\mathrm { f } ( x ) = x ^ { 2 }\) for \(0 \leq x \leq 1\)
It is also given that \(\mathrm { f } ( x )\) is a linear function for \(1 < x \leq \frac { 3 } { 2 }\)
For all other values of \(x , \mathrm { f } ( x ) = 0\) A sketch of the graph of \(y = \mathrm { f } ( x )\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{c309e27b-5618-4f94-aecd-a55d8756ef03-12_821_1077_758_543} Show that \(\operatorname { Var } ( X ) = 0.0864\) correct to three significant figures.
\includegraphics[max width=\textwidth, alt={}, center]{c309e27b-5618-4f94-aecd-a55d8756ef03-14_2491_1755_173_123} Additional page, if required. number Write the question numbers in the left-hand margin.
AQA Further AS Paper 2 Statistics 2024 June Q1
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
AQA Further AS Paper 2 Statistics 2024 June Q2
1 marks
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.
ObservedXYZ
A28666
B884
C541610
Expected\(\mathbf { X }\)\(\mathbf { Y }\)\(\mathbf { Z }\)
\(\mathbf { A }\)451540
\(\mathbf { B }\)938
\(\mathbf { C }\)361232
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
AQA Further AS Paper 2 Statistics 2024 June Q3
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.
AQA Further AS Paper 2 Statistics 2024 June Q4
4 marks
4 The discrete random variable \(Y\) has probability distribution
\(y\)15213643
\(\mathrm { P } ( Y = y )\)0.160.320.290.23
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 \(\mathrm { P } ( Y > m - 1.5 s )\)
AQA Further AS Paper 2 Statistics 2024 June Q5
1 marks
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
    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
    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
    		12345678
    Frequency376411216130815610953
    5
    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.
AQA Further AS Paper 2 Statistics 2024 June Q6
2 marks
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.
AQA Further AS Paper 2 Statistics 2024 June Q7
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.
    DO NOT WRITE/ON THIS PAGE ANSWER IN THE/SPACES PROVIDED number Additional page, if required. Write the question numbers in the left-hand margin.
    Additional page, if required. number Additional page, if required.
    Write the question numbers in the left-hand margin. Additional page, if required. number Additional page, if required.
    Write the question numbers in the left-hand margin.
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