Questions — AQA Further Paper 3 Statistics (56 questions)

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AQA Further Paper 3 Statistics 2021 June Q2
1 marks
2 The random variable \(X\) has variance \(\operatorname { Var } ( X )\) Which of the following expressions is equal to \(\operatorname { Var } ( a X + b )\), where \(a\) and \(b\) are non-zero constants? Circle your answer.
[0pt] [1 mark]
\(a \operatorname { Var } ( X )\)
\(a \operatorname { Var } ( X ) + b\)
\(a ^ { 2 } \operatorname { Var } ( X )\)
\(a ^ { 2 } \operatorname { Var } ( X ) + b\)
AQA Further Paper 3 Statistics 2021 June Q3
3 In a game, it is only possible to score 10, 20 or 30 points. The probability of scoring 20 points is twice the probability of scoring 30 points.
The probability of scoring 20 points is half the probability of scoring 10 points.
3
  1. Find the mean points scored when the game is played once, giving your answer to two decimal places.
    3
  2. Mina plays the game.
    Her father, Michael, tells her that he will multiply her score by 5 and then subtract 10 He will then give her the value he has calculated in pence rounded to the nearest penny. Calculate the expected value in pence that Mina receives.
AQA Further Paper 3 Statistics 2021 June Q4
4 Oscar is studying the daily maximum temperature in \({ } ^ { \circ } \mathrm { C }\) in a village during the month of June. He constructs a \(95 \%\) confidence interval of width \(0.8 ^ { \circ } \mathrm { C }\) using a random sample of 150 days. He assumes that the daily maximum temperature has a normal distribution.
4
  1. Find the standard deviation of Oscar's sample, giving your answer to three significant figures.
    4
  2. Oscar calculates the mean of his sample to be \(25.3 ^ { \circ } \mathrm { C }\)
    He claims that the population mean is \(26.0 ^ { \circ } \mathrm { C }\)
    Explain whether or not his confidence interval supports his claim.
    4
  3. Explain how Oscar could reduce the width of his 95\% confidence interval.
AQA Further Paper 3 Statistics 2021 June Q5
5 The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c l } 0 & x \leq 1
\frac { 1 } { 10 } x - \frac { 1 } { 10 } & 1 < x \leq 6
\frac { 1 } { 90 } x ^ { 2 } + \frac { 1 } { 10 } & 6 < x \leq 9
1 & x > 9 \end{array} \right.$$ 5
  1. Find the probability density function \(\mathrm { f } ( x )\)
    5
  2. Show that \(\operatorname { Var } ( X ) = \frac { 6737 } { 1200 }\)
    \includegraphics[max width=\textwidth, alt={}, center]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-07_2488_1716_219_153}
AQA Further Paper 3 Statistics 2021 June Q6
6 Danai is investigating the number of speeding offences in different towns in a country. She carries out a hypothesis test to test for association between town and number of speeding offences per year. 6
  1. State the hypotheses for this test. 6
  2. The observed frequencies, \(O\), have been collected and the expected frequencies, \(E\), have been calculated in an \(n \times m\) contingency table, where \(n > 3\) and \(m > 3\) One of the values of \(E\) is less than 5 6
    1. Explain what steps Danai should take before calculating the test statistic.
      6
  4. (ii) State an expression for the test statistic Danai should calculate.
    6
  5. Danai correctly calculates the value of the test statistic to be 45.22 The number of degrees of freedom for the test is 25
    Determine the outcome of Danai's test, using the \(1 \%\) level of significance.
AQA Further Paper 3 Statistics 2021 June Q7
7 The random variable \(X\) has an exponential distribution with parameter \(\lambda\) 7
  1. Prove that \(\mathrm { E } ( X ) = \frac { 1 } { \lambda }\)
    7
  2. Prove that \(\operatorname { Var } ( X ) = \frac { 1 } { \lambda ^ { 2 } }\)
AQA Further Paper 3 Statistics 2021 June Q8
8 A company records the number of complaints, \(X\), that it receives over 60 months. The summarised results are $$\sum x = 102 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 103.25$$ 8
  1. Using this data, explain why it may be appropriate to model the number of complaints received by the company per month by a Poisson distribution with mean 1.7
    8
  2. The company also receives enquiries as well as complaints. The number of enquiries received is independent of the number of complaints received. The company models the number of complaints per month with a Poisson distribution with mean 1.7 and the number of enquiries per month with a Poisson distribution with mean 5.2 The company starts selling a new product.
    The company records a total of 3 complaints and enquiries in one randomly chosen month. Investigate if the mean total number of complaints and enquiries received per month has changed following the introduction of the new product, using the \(10 \%\) level of significance.
    8
  3. It is later found that the mean total number of complaints and enquiries received per month is 6.1 Find the power of the test carried out in part (b), giving your answer to four decimal places.
    \includegraphics[max width=\textwidth, alt={}, center]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-15_2492_1721_217_150}
    \includegraphics[max width=\textwidth, alt={}]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-20_2496_1723_214_148}
AQA Further Paper 3 Statistics 2022 June Q1
1 The random variable \(T\) follows a discrete uniform distribution and can take values \(1,2,3 , \ldots , 16\) Find the variance of \(T\) Circle your answer.
1.2518 .7521 .2521 .33
AQA Further Paper 3 Statistics 2022 June Q2
1 marks
2 The random variable \(X\) has probability density function $$f ( x ) = \begin{cases} 1 & 0 < x \leq \frac { 1 } { 2 }
\frac { 3 } { 8 } x ^ { - 2 } & \frac { 1 } { 2 } < x \leq \frac { 3 } { 2 }
0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X < 1 )\) Circle your answer.
[0pt] [1 mark]
\(\frac { 1 } { 8 }\)
\(\frac { 3 } { 8 }\)
\(\frac { 5 } { 8 }\)
\(\frac { 7 } { 8 }\)
\includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-03_2488_1718_219_153}
AQA Further Paper 3 Statistics 2022 June Q3
3 marks
3 The random variable \(X\) has an exponential distribution with probability density function \(\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x }\) where \(x \geq 0\) 3
  1. Show that the cumulative distribution function, for \(x \geq 0\), is given by \(\mathrm { F } ( x ) = 1 - \mathrm { e } ^ { - \lambda x }\)
    [0pt] [3 marks]
    3
  2. Given that \(\lambda = 2\), find \(\mathrm { P } ( X > 1 )\), giving your answer to three decimal places.
AQA Further Paper 3 Statistics 2022 June Q4
4 Daisies and dandelions are the only flowers growing in a field. The number of daisies per square metre in the field has a mean of 16
The number of dandelions per square metre in the field has a mean of 10
The number of daisies per square metre and the number of dandelions per square metre are independent. 4
  1. Using a Poisson model, find the probability that a randomly selected square metre from the field has a total of at least 30 flowers, giving your answer to three decimal places.
    4
  2. A survey of the entire field is taken.
    The standard deviation of the total number of flowers per square metre is 10 State, with a reason, whether the model used in part (a) is valid.
AQA Further Paper 3 Statistics 2022 June Q5
5 The mass, \(X\), in grams of a particular type of apple is modelled using a normal distribution. A random sample of 12 apples is collected and the summarised results are $$\sum x = 1038 \quad \text { and } \quad \sum x ^ { 2 } = 90100$$ 5
  1. A 99\% confidence interval for the population mean of the masses of the apples is constructed using the random sample. Show that the confidence interval is \(( 81.7,91.3 )\) with values correct to three significant figures.
    5
  2. Padraig claims that the population mean mass of the apples is 85 grams. He carries out a hypothesis test at the \(1 \%\) level of significance using the random sample of 12 apples. The hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 85
    & \mathrm { H } _ { 1 } : \mu \neq 85 \end{aligned}$$ State, with a reason, whether the null hypothesis is accepted or rejected.
    5
  3. Interpret, in context, the conclusion to the hypothesis test in part (b).
AQA Further Paper 3 Statistics 2022 June Q6
2 marks
6 The discrete random variable \(X\) has probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 0
b & x = 1
c & x = 2
0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The mean of \(X\) is 1.2 and the variance of \(X\) is 0.56
6
  1. Deduce the values of \(a , b\) and \(c\)
    6
  2. The continuous random variable \(Y\) is independent of \(X\) and has variance 15 Find \(\operatorname { Var } ( X - 2 Y - 11 )\)
    [0pt] [2 marks]
AQA Further Paper 3 Statistics 2022 June Q7
1 marks
7
  1. Test the scientist's claim, using the 10\% level of significance.
    7
  2. For the context of the test carried out in part (a), state the meaning of a Type I error. [1 mark]
AQA Further Paper 3 Statistics 2022 June Q8
2 marks
8 The continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) where $$\mathrm { F } ( x ) = \begin{cases} 0 & x = 0
\mathrm { e } ^ { k x } - 1 & 0 \leq x \leq 5
1 & x > 5 \end{cases}$$ 8
  1. Show that \(k = \frac { 1 } { 5 } \ln 2\)
    [0pt] [2 marks]
    8
  2. Show that the median of \(X\) is \(a \frac { \ln b } { \ln 2 } - c\), where \(a , b\) and \(c\) are integers to be found.
    8
  		3. Show that the mean of \(X\) is \(p - \frac { q } { \ln 2 }\), where \(p\) and \(q\) are integers to be found.
AQA Further Paper 3 Statistics 2022 June Q9
2 marks
9 Lianne models the maximum time in hours that a rechargeable battery can be used, before needing to be recharged, with a rectangular distribution with values between 8 and 12 9
  1. The probability that the maximum time the battery can be used before needing to be recharged is more than 10.5 hours is equal to \(p\) Lianne will only buy the battery if \(p\) is more than 0.4
    Determine whether Lianne will buy the battery.
    [0pt] [2 marks]
    9
  2. A histogram is plotted for 100 recharges showing the maximum time the battery can be used before needing to be recharged.
    \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-15_670_1186_404_427} Explain why the model used in part (a) may not be valid and suggest the name of a different distribution that could be used to model the maximum time between recharges.
    \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-16_2488_1732_219_139}
    \includegraphics[max width=\textwidth, alt={}]{62cee897-6eac-40b3-84c1-a0d165ba6903-20_2496_1721_214_148}
AQA Further Paper 3 Statistics 2023 June Q1
1 The discrete random variable \(A\) takes only the values 0,2 and 4, and has cumulative distribution function \(\mathrm { F } ( a ) = \mathrm { P } ( A \leq a )\)
\(a\)024
\(\mathrm {~F} ( a )\)0.20.61
Find \(\mathrm { P } ( A = 2 )\)
Circle your answer. \(0 \quad 0.4 \quad 0.6 \quad 0.8\)
AQA Further Paper 3 Statistics 2023 June Q2
1 marks
2 The time, \(T\) days, between rain showers in a city in autumn can be modelled by an exponential distribution with mean 1.25 Find the distribution of the number of rain showers per day in the city.
Tick ( \(\checkmark\) ) one box.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-03_108_113_1800_370}
DistributionMean
Exponential0.8
\includegraphics[max width=\textwidth, alt={}]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-03_108_113_1932_370}
Exponential1.25
□
Poisson0.8
□
Poisson1.25
AQA Further Paper 3 Statistics 2023 June Q3
3 The masses of tins of a particular brand of spaghetti are normally distributed with mean \(\mu\) grams and standard deviation 4.1 grams. A random sample of 11 tins of spaghetti has a mean mass of 401.8 grams.
Construct a \(98 \%\) confidence interval for \(\mu\), giving your values to one decimal place.
AQA Further Paper 3 Statistics 2023 June Q4
4 The random variable \(X\) has a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\) A random sample of 8 observations of \(X\) has mean \(\bar { x } = 101.5\) and gives the unbiased estimate of the variance as \(s ^ { 2 } = 4.8\) The random sample is used to conduct a hypothesis test at the \(10 \%\) level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 100
& \mathrm { H } _ { 1 } : \mu \neq 100 \end{aligned}$$ Carry out the hypothesis test.
AQA Further Paper 3 Statistics 2023 June Q5
5 A school management team oversees 11 different schools.
The school management team allows each student in the schools to choose one enrichment activity from 11 possible activities. The school management team count the number of students in each school choosing each of the possible activities. They then conduct a \(\chi ^ { 2 }\)-test for association with the data they have gathered. 5
  1. Exactly one of the calculated expected frequencies for the \(\chi ^ { 2 }\)-test is less than 5
    Explain why the number of degrees of freedom for the test is 90
    5
  2. The school management team claims that there is an association between the school a student attends and the activity they choose. The test statistic is 124.8 Test the claim using the \(1 \%\) level of significance.
    5
  3. During the hypothesis test, the value of \(\frac { ( O - E ) ^ { 2 } } { E }\), where \(O\) is the observed frequency and \(E\) is the expected frequency, was calculated for each group of students. The values for four groups of students are shown in the table below.
    Group\(\frac { ( O - E ) ^ { 2 } } { E }\)
    Attends school 3 and chose activity 10.01
    Attends school 8 and chose activity 318.5
    Attends school 8 and chose activity 724.2
    Attends school 11 and chose activity 749.0
    State, with a reason, which of the four groups of students represents the strongest source of association.
AQA Further Paper 3 Statistics 2023 June Q6
6 A game consists of two rounds. The first round of the game uses a random number generator to output the score \(X\), a real number between 0 and 10 6
  1. Find \(\mathrm { P } ( X > 4 )\) 6
  2. The second round of the game uses an unbiased dice, with faces numbered 1 to 6 , to give the score \(Y\) The variables \(X\) and \(Y\) are independent.
    6
    1. Find the mean total score of the game.
      6
  4. (ii) Find the variance of the total score of the game.
AQA Further Paper 3 Statistics 2023 June Q7
7 Company \(A\) uses a machine to produce toys. The number of toys in a week that do not pass Company \(A\) 's quality checks is modelled by a Poisson distribution \(X\) with standard deviation 5 The machine producing the toys breaks down.
After it is repaired, 16 toys in the next week do not pass the quality checks.
7
  1. Investigate whether the average number of toys that do not pass the quality checks in a week has changed, using the \(5 \%\) level of significance.
    7
  2. For the test carried out in part (a), state in context the meaning of a Type II error. 7
  3. Company \(B\) uses a different machine to produce toys.
    The number of toys in a week that do not pass Company B's quality checks is modelled by a Poisson distribution \(Y\) with mean 18 The variables \(X\) and \(Y\) are independent.
    Find the distribution of the total number of toys in a week produced by companies \(A\) and \(B\) that do not pass their quality checks. 7
  4. State two reasons why a Poisson distribution may not be a valid model for the number of toys that do not pass the quality checks in a week. Reason 1 \(\_\_\_\_\)
    Reason 2 \(\_\_\_\_\)
AQA Further Paper 3 Statistics 2023 June Q8
8 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k \sin 2 x & 0 \leq x \leq \frac { \pi } { 6 }
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. 8
  1. Show that \(k = 4\)
    8
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\)
    8
  3. Find the median of \(X\), giving your answer to three significant figures. 8
  4. Find the mean of \(X\) giving your answer in the form \(\frac { 1 } { a } ( b \sqrt { 3 } - \pi )\) where \(a\) and \(b\) are integers.
    \includegraphics[max width=\textwidth, alt={}, center]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-14_2492_1721_217_150}
AQA Further Paper 3 Statistics 2024 June Q1
1 The random variable \(X\) has a Poisson distribution with mean 16 Find the standard deviation of \(X\) Circle your answer.
4
8
16
256