Questions Further AS Paper 2 Statistics (60 questions)

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AQA Further AS Paper 2 Statistics 2019 June Q3
5 marks Moderate -0.8
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.
% \(\begin{aligned4 \text { The continuous random variable } X \text { has probability density fu }
\qquad f ( x ) = \begin{cases} \frac { 4 } { 99 } \left( 12 x - x ^ { 2 } - x ^ { 3 } \right)0 \leq x \leq 3
0\text { otherwise } \end{cases} \end{aligned}\)}
AQA Further AS Paper 2 Statistics 2019 June Q4
8 marks Standard +0.3
4
  1. \(\text { Find } \mathrm { P } ( X > 1 )\) [0pt] [3 marks]
    4

  2. [0pt] [3 marks]
    □
    4
  3. Find \(\mathrm { E } \left( 2 X ^ { - 1 } - 3 \right)\)
AQA Further AS Paper 2 Statistics 2019 June Q5
9 marks Standard +0.8
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. Prove that \(\mathrm { E } ( X ) = \frac { n + 1 } { 2 }\) [0pt] [3 marks]
      5
      1. (ii) Prove that \(\operatorname { Var } ( X ) = \frac { n ^ { 2 } - 1 } { 12 }\)
        5
      2. State two conditions under which a discrete uniform distribution can be used to model the score when a cubic dice is rolled.
        [2 marks]
AQA Further AS Paper 2 Statistics 2019 June Q6
7 marks Standard +0.3
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.
AQA Further AS Paper 2 Statistics 2019 June Q7
9 marks Standard +0.3
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:
AQA Further AS Paper 2 Statistics 2022 June Q1
1 marks Easy -2.0
1 The discrete random variable \(X\) has the following probability distribution
\(x\)- 151829
\(\mathrm { P } ( X = x )\)0.20.70.1
Find \(\mathrm { P } ( X > 18 )\) Circle your answer.
0.1
0.2
0.7
0.8
AQA Further AS Paper 2 Statistics 2022 June Q2
1 marks Moderate -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
AQA Further AS Paper 2 Statistics 2022 June Q3
7 marks Easy -1.2
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 \(\operatorname { Var } ( 9 A - 2 )\)
AQA Further AS Paper 2 Statistics 2022 June Q4
4 marks Moderate -0.3
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.
AQA Further AS Paper 2 Statistics 2022 June Q5
11 marks Standard +0.3
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 (d)5
  3. Show that \(\mathrm { E } \left( \frac { 1 } { X ^ { 2 } } \right) = \frac { 133 } { 212 }\)
AQA Further AS Paper 2 Statistics 2022 June Q6
8 marks Standard +0.3
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.
AQA Further AS Paper 2 Statistics 2022 June Q7
8 marks Standard +0.3
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.
AQA Further AS Paper 2 Statistics 2023 June Q1
1 marks Easy -1.2
1 The continuous random variable \(X\) has variance 9 The discrete random variable \(Y\) has standard deviation 2 and is independent of \(X\) Find \(\operatorname { Var } ( X + Y )\) Circle your answer.
5111385
AQA Further AS Paper 2 Statistics 2023 June Q2
1 marks Easy -1.2
2 The random variable \(T\) has a discrete uniform distribution and takes the values 1, 2, 3, 4 and 5 Find the variance of \(T\) Circle your answer.
\(\frac { 1 } { 5 }\)\(\frac { 4 } { 3 }\)2\(\frac { 13 } { 6 }\)
AQA Further AS Paper 2 Statistics 2023 June Q3
3 marks Easy -1.2
3 The discrete random variable \(X\) has probability distribution
\(x\)- 438
\(\mathrm { P } ( X = x )\)0.20.70.1
Show that \(\mathrm { E } ( 5 X - 7 ) = 3.5\)
AQA Further AS Paper 2 Statistics 2023 June Q4
4 marks Standard +0.3
4 The proportion, \(p\), of people in a particular town who use the local supermarket is unknown. A random sample of 30 people in the town is taken and each person is asked if they use the local supermarket. The manager of the supermarket claims that 35\% of the people in the town use the local supermarket. The random sample is used to conduct a hypothesis test at the \(5 \%\) level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : p = 0.35 \\ & \mathrm { H } _ { 1 } : p \neq 0.35 \end{aligned}$$ Show that the probability that a Type I error is made is 0.0356 , correct to four decimal places.
AQA Further AS Paper 2 Statistics 2023 June Q5
6 marks Moderate -0.3
5 Rebekah is investigating the distances, \(X\) light years, between the Earth and visible stars in the night sky. She determines the distance between the Earth and a star for a random sample of 100 visible stars. The summarised results are as follows: $$\sum x = 35522 \quad \text { and } \quad \sum x ^ { 2 } = 32902257$$ 5
  1. Calculate a 97\% confidence interval for the population mean of \(X\), giving your values to the nearest light year.
    5
  2. Mike claims that the population mean is 267 light years. Rebekah says that the confidence interval supports Mike's claim. State, with a reason, whether Rebekah is correct.
AQA Further AS Paper 2 Statistics 2023 June Q6
8 marks Standard +0.3
6 An insurance company models the number of motor claims received in 1 day using a Poisson distribution with mean 65 6
  1. Find the probability that the company receives at most 60 motor claims in 1 day. Give your answer to three decimal places. 6
  2. The company receives motor claims using a telephone line which is open 24 hours a day. Find the probability that the company receives exactly 2 motor claims in 1 hour. Give your answer to three decimal places.
    6
  3. The company models the number of property claims received in 1 day using a Poisson distribution with mean 23 Assume that the number of property claims received is independent of the number of motor claims received. 6 (c) (i) Find the standard deviation of the variable that represents the total number of motor claims and property claims received in 1 day. Give your answer to three significant figures.
    6 (c) (ii) Find the probability that the company receives a total of more than 90 motor claims and property claims in 1 day. Give your answer to three significant figures.
AQA Further AS Paper 2 Statistics 2023 June Q7
10 marks Standard +0.3
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 marks Challenging +1.2
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 marks Easy -2.0
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 Easy -1.2
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 marks Moderate -0.8
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
7 marks Easy -1.2
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
6 marks Easy -1.8
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. (i) State a distribution that could be used to model the number that the spinner lands on. 5
    2. (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
    4. 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
    6. (ii) Describe how Clare's results could be used to adjust the model.