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

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA Further AS Paper 2 Statistics 2019 June Q3
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} \(\begin{aligned} & 4 \text { The continuous random variable } X \text { has probability density fu }
& \qquad f ( x ) = \begin{cases} \frac { 4 } { 99 } \left( 12 x - x ^ { 2 } - x ^ { 3 } \right) & 0 \leq x \leq 3
0 & \text { otherwise } \end{cases} \end{aligned}\)
AQA Further AS Paper 2 Statistics 2019 June Q4
6 marks
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 \(\mathrm { E } \left( 2 X ^ { - 1 } - 3 \right)\)
AQA Further AS Paper 2 Statistics 2019 June Q5
5 marks
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
  2. (ii) Prove that \(\operatorname { Var } ( X ) = \frac { n ^ { 2 } - 1 } { 12 }\)
    5
  3. 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
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
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
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
2 marks
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
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
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
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
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
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 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
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
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 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
3 marks
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
6 marks
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
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 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
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 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 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
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
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.