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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 2020 June Q1
1 marks Easy -2.0
1 The discrete random variable \(X\) has the following probability distribution function. $$\mathrm { P } ( X = x ) = \begin{cases} 0.2 & x = 1 \\ 0.3 & x = 2 \\ 0.1 & x = 3,4 \\ 0.25 & x = 5 \\ 0.05 & x = 6 \\ 0 & \text { otherwise } \end{cases}$$ Find the mode of \(X\). Circle your answer.
[0pt] [1 mark]
0.10 .2523 \(2 \quad \mathrm {~A} \chi ^ { 2 }\) test is carried out in a school to test for association between the class a student belongs to and the number of times they are late to school in a week. The contingency table below gives the expected values for the test.
Number of times late
\cline { 2 - 7 }\cline { 2 - 6 }\(\mathbf { 0 }\)\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)
\cline { 2 - 7 }\(\mathbf { A }\)8.121415.12144.76
\cline { 2 - 7 } Class\(\mathbf { B }\)8.9915.516.7415.55.27
\cline { 2 - 7 }\(\mathbf { C }\)11.8920.522.1420.56.97
Find a possible value for the degrees of freedom for the test. Circle your answer. 681215
AQA Further AS Paper 2 Statistics 2020 June Q3
5 marks Moderate -0.8
3 The random variable \(X\) represents the value on the upper face of an eight-sided dice after it has been rolled. The faces are numbered 1 to 8 The random variable \(X\) is modelled by a discrete uniform distribution with \(n = 8\) 3
  1. Find \(\mathrm { E } ( X )\) 3
  2. \(\quad\) Find \(\operatorname { Var } ( X )\) 3
  3. Find \(\mathrm { P } ( X \geq 6 )\) 3
  4. The dice was rolled 800 times and the results below were obtained.
    \(\boldsymbol { x }\)\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)\(\mathbf { 5 }\)\(\mathbf { 6 }\)\(\mathbf { 7 }\)\(\mathbf { 8 }\)
    Frequency1036384110744185240
    State, with a reason, how you would refine the model for the random variable \(X\).
    [0pt] [2 marks]
AQA Further AS Paper 2 Statistics 2020 June Q4
3 marks Moderate -0.8
4 Murni is investigating the annual salary of people from a particular town. She takes a random sample of 200 people from the town and records their annual salary. The mean annual salary is \(\pounds 28500\) and the standard deviation is \(\pounds 5100\) Calculate a \(97 \%\) confidence interval for the population mean of annual salaries for the people who live in the town, giving your values to the nearest pound. \includegraphics[max width=\textwidth, alt={}, center]{0d592978-08eb-40a8-ab3b-88339956b89d-07_2488_1716_219_153}
AQA Further AS Paper 2 Statistics 2020 June Q5
7 marks Moderate -0.8
5 The discrete random variable \(X\) has the following probability distribution.
\(\boldsymbol { x }\)\(\mathbf { 2 }\)\(\mathbf { 4 }\)\(\mathbf { 6 }\)\(\mathbf { 9 }\)
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.20.60.10.1
5
  1. Find \(\mathrm { P } ( X \leq 6 )\) 5
  2. Let \(Y = 3 X + 2\) Show that \(\operatorname { Var } ( Y ) = 32.49\) 5
  3. The continuous random variable \(T\) is independent of \(Y\). Given that \(\operatorname { Var } ( T ) = 5\), find \(\operatorname { Var } ( T + Y )\)
AQA Further AS Paper 2 Statistics 2020 June Q6
8 marks Standard +0.3
6 The continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 4 } { 45 } \left( x ^ { 3 } - 10 x ^ { 2 } + 29 x - 20 \right) & 1 \leq x \leq 4 \\ 0 & \text { otherwise } \end{array} \right.$$ 6
  1. Find \(\mathrm { P } ( X < 2 )\) 6
  2. Verify that the median of \(X\) is 2.3 , correct to two significant figures.
    6
  3. Find the mean of \(X\).
AQA Further AS Paper 2 Statistics 2020 June Q7
6 marks Moderate -0.8
7 A restaurant has asked Sylvia to conduct a \(\chi ^ { 2 }\) test for association between meal ordered and age of customer. 7
  1. State the hypotheses that Sylvia should use for her test. 7
  2. Sylvia correctly calculates her value of the test statistic to be 44.1
    She uses a \(5 \%\) level of significance and the degrees of freedom for the test is 30
    Sylvia accepts the null hypothesis.
    Explain whether or not Sylvia was correct to accept the null hypothesis.
    7
  3. State in context the correct conclusion to Sylvia's test.
AQA Further AS Paper 2 Statistics 2020 June Q8
9 marks Standard +0.8
8 There are two hospitals in a city. Over a period of time, the first hospital recorded an average of 20 births a day.
Over the same period of time, the second hospital recorded an average of 5 births a day. Stuart claims that birth rates in the hospitals have changed over time.
On a randomly chosen day, he records a total of 16 births from the two hospitals.
8
  1. Investigate Stuart's claim, using a suitable test at the \(5 \%\) level of significance.
    8
  2. For a test of the type carried out in part (a), find the probability of making a Type I error, giving your answer to two significant figures. \includegraphics[max width=\textwidth, alt={}, center]{0d592978-08eb-40a8-ab3b-88339956b89d-16_2490_1735_219_139}
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
  		3. 5
  4. 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
    1. 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
  5. (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