Questions SPS SM Statistics (77 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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
SPS SPS SM Statistics 2021 May Q1
  1. Three Bags, \(A , B\) and \(C\), each contain 1 red marble and some green marbles.
\begin{displayquote} Bag \(A\) contains 1 red marble and 9 green marbles only
Bag \(B\) contains 1 red marble and 4 green marbles only
Bag \(C\) contains 1 red marble and 2 green marbles only \end{displayquote} Sasha selects at random one marble from \(\operatorname { Bag } A\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from Bag \(B\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from Bag \(C\).
  1. Draw a tree diagram to represent this information.
  2. Find the probability that Sasha selects 3 green marbles.
  3. Find the probability that Sasha selects at least 1 marble of each colour.
  4. Given that Sasha selects a red marble, find the probability that he selects it from Bag \(B\).
SPS SPS SM Statistics 2021 May Q2
2.
  1. The variable \(X\) has the distribution \(\mathrm { N } ( 20,9 )\).
    (a) Find \(\mathrm { P } ( X > 25 )\).
    (b) Given that \(\mathrm { P } ( X > a ) = 0.2\), find \(a\).
    (c) Find \(b\) such that \(\mathrm { P } ( 20 - b < X < 20 + b ) = 0.5\).
  2. The variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \frac { \mu ^ { 2 } } { 9 } \right)\). Find \(\mathrm { P } ( Y > 1.5 \mu )\).
SPS SPS SM Statistics 2021 May Q3
3. Laxmi wishes to test whether there is linear correlation between the mass and the height of adult males.
  1. State, with a reason, whether Laxmi should use a 1-tail or a 2-tail test. Laxmi chooses a random sample of 40 adult males and calculates Pearson's product-moment correlation coefficient, \(r\). She finds that \(r = 0.2705\).
  2. Use the table below to carry out the test at the \(5 \%\) significance level. Critical values of Pearson's product-moment correlation coefficient.
    \cline{2-5}
    1-tail
    test
    		\(5 \%\)\(2.5 \%\)\(1 \%\)
    2-tail
    test
    		\(10 \%\)\(5 \%\)\(2.5 \%\)\(1 \%\)
    380.27090.32020.37600.4128
    390.26730.31600.37120.4076
    400.26380.31200.36650.4026
    410.26050.30810.36210.3978
SPS SPS SM Statistics 2021 May Q4
  1. A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, \(D \mathrm { ml }\), follows a normal distribution with mean 25 ml
Given that \(15 \%\) of bottles contain less than 24.63 ml
  1. find, to 2 decimal places, the value of \(k\) such that \(\mathrm { P } ( 24.63 < D < k ) = 0.45\) A random sample of 200 bottles is taken.
  2. Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and \(k \mathrm { ml }\)
SPS SPS SM Statistics 2021 May Q5
5. Two events C and D are such that \(P ( C \mid D ) = 3 \times P ( C )\) where \(P ( C ) \neq 0\).
  1. Explain whether or not events C and D could be independent events. Given also that $$P ( C \cap D ) = \frac { 1 } { 2 } \times P ( C ) \text { and } P \left( C ^ { \prime } \cap D ^ { \prime } \right) = \frac { 7 } { 10 }$$
  2. find \(P ( C )\), showing your working clearly.
SPS SPS SM Statistics 2021 May Q6
6. The level, in grams per millilitre, of a pollutant at different locations in a certain river is denoted by the random variable \(X\), where \(X\) has the distribution \(\mathrm { N } ( \mu , 0.0000409 )\). In the past the value of \(\mu\) has been 0.0340 .
This year the mean level of the pollutant at 50 randomly chosen locations was found to be 0.0325 grams per millilitre. Test, at the \(5 \%\) significance level, whether the mean level of pollutant has changed.
SPS SPS SM Statistics 2021 September Q1
  1. A random sample of distances travelled to work for 120 commuters from a train station in Devon is recorded. The distances travelled, to the nearest mile, are summarised below.
Distance (to the nearest mile)Number of commuters
0-910
10-1919
20-2943
30-3925
40-498
50-596
60-695
70-793
80-891
For this distribution:
a estimate the median. The mid-point of each class was represented by \(x\) and its corresponding frequency by \(f\). The mid-point of the lowest class was taken to be 4.75 giving: $$\Sigma f x = 3552.5 \text { and } \Sigma f x ^ { 2 } = 138043.125$$ b Estimate the mean and the standard deviation of this distribution.
c Explain why the median is less than the mean for these data.
[0pt] [BLANK PAGE]
SPS SPS SM Statistics 2021 September Q2
2. Data relating to the lifetimes (to the nearest hour) of a random sample of 200 light bulbs from the production line of a manufacturer were summarised in a grouped frequency table. The mid-point of each class in the table was represented by \(x\) and the corresponding frequency for that class by \(f\). The data were then coded using: $$y = \frac { ( x - 755.0 ) } { 2.5 }$$ and summarised as follows: $$\sum f y = - 467 , \sum f y ^ { 2 } = 9179$$ Calculate estimates of the mean and the standard deviation of the lifetimes of this sample of bulbs.
(4 marks)
[0pt] [BLANK PAGE]
SPS SPS SM Statistics 2021 September Q3
3. There are 180 students at a college following a general course in computing. Students on this course can choose to take up to three extra options. \begin{displayquote} 112 take systems support,
70 take developing software,
81 take networking,
35 take developing software and systems support,
28 take networking and developing software,
40 take systems support and networking,
4 take all three extra options. \end{displayquote} a Draw a Venn diagram to represent this information. A student from the course is chosen at random.
b Find the probability that this student takes
i none of the three extra options
ii networking only. Students who take systems support and networking are eligible to become technicians.
c Given that the randomly chosen student is eligible to become a technician, find the probability that this student takes all three extra options.
[0pt] [BLANK PAGE]
SPS SPS SM Statistics 2021 September Q4
4. A company assembles drills using components from two sources. Goodbuy supplies the components for \(85 \%\) of the drills whilst Amart supplies the components for the rest.
It is known that \(3 \%\) of the components supplied by Goodbuy are faulty and \(6 \%\) of those supplied by Amart are faulty.
a Represent this information on a tree diagram. An assembled drill is selected at random.
b Find the probability that the drill is not faulty.
SPS SPS SM Statistics 2021 September Q5
5. Figure 2 is a histogram showing the distribution of the time taken in minutes, \(t\), by a group of people to swim 500 m . \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{a58a2c41-3a53-41cc-b80f-84adb04a5f5c-11_547_1120_333_374}
\end{figure} a Find the probability that a person chosen at random from the group takes longer than 18 minutes.
SPS SPS SM Statistics 2021 September Q6
6. The discrete random variable \(X\) has probability function
\(\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) & x = 0,1,2
k ( x - 2 ) & x = 3
0 & \text { otherwise } \end{cases}\)
where \(k\) is a positive constant.
a Show that \(k = 0.25\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
b Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\)
c Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
d Find \(\mathrm { P } \left( 1.3 \leqslant X _ { 1 } + X _ { 2 } \leqslant 3.2 \right)\)
SPS SPS SM Statistics 2021 September Q7
7. Emma throws a fair coin 15 times and records the number of times it shows a head.
a State the appropriate distribution to model the number of times the coin shows a head giving any relevant parameter values.
b Find the probability that Emma records:
i exactly 8 heads
ii at least 4 heads.
SPS SPS SM Statistics 2022 February Q1
1. Answer all the questions. $$f ( x ) = 3 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 10$$
  1. Use the factor theorem to show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\)
  2. Find the values of the constants \(a , b\) and \(c\) such that $$\mathrm { f } ( x ) \equiv ( x - 2 ) \left( a x ^ { 2 } + b x + c \right)$$
  3. Using your answer to part (b) show that the equation \(\mathrm { f } ( x ) = 0\) has only one real root.
    [0pt] [BLANK PAGE]
SPS SPS SM Statistics 2022 February Q2
2. \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.}
  1. Express as an integral $$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 12 } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \delta x$$
  2. Using your answer to part (a) show that $$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 12 } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \delta x = \frac { 98 } { 3 }$$ [BLANK PAGE]
SPS SPS SM Statistics 2022 February Q3
3. The curve \(C\) has equation $$y = 5 x ^ { 4 } - 24 x ^ { 3 } + 42 x ^ { 2 } - 32 x + 11 \quad x \in \mathbb { R }$$
  1. Find
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
    1. Verify that \(C\) has a stationary point at \(x = 1\)
    2. Show that this stationary point is a point of inflection, giving reasons for your answer.
      [0pt] [BLANK PAGE]
SPS SPS SM Statistics 2022 February Q4
4. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l l } \mathrm { f } ( x ) = \frac { k x } { 2 x - 1 } & x \in \mathbb { R } & x \neq \frac { 1 } { 2 }
\mathrm {~g} ( x ) = 2 + 3 x - x ^ { 2 } & x \in \mathbb { R } & \end{array}$$ where \(k\) is a non-zero constant.
  1. Find in terms of \(k\)
    1. \(\mathrm { fg } ( 4 )\)
    2. the range of f
    3. \(\mathrm { f } ^ { - 1 }\) Given that $$\mathrm { f } ^ { - 1 } ( 2 ) = \frac { 11 } { 3 \mathrm {~g} ( 2 ) }$$
  2. find the exact value of \(k\)
    [0pt] [BLANK PAGE]
SPS SPS SM Statistics 2022 February Q5
5. $$f ( x ) = \frac { 10 } { \sqrt { 4 - 3 x } }$$
  1. Show that the first 4 terms in the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), are $$A + B x + C x ^ { 2 } + \frac { 675 } { 1024 } x ^ { 3 }$$ where \(A , B\) and \(C\) are constants to be found. Give each constant in simplest form. Given that this expansion is valid for \(| x | < k\)
  2. state the largest value of \(k\). By substituting \(x = \frac { 1 } { 3 }\) into \(\mathrm { f } ( x )\) and into the answer for part (a),
  3. find an approximation for \(\sqrt { 3 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers to be found.
    [0pt] [BLANK PAGE]
SPS SPS SM Statistics 2022 February Q6
6. \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.}
  1. Express \(3 \cos x + \sin x\) in the form \(R \cos ( x - \alpha )\) where
    • \(\quad R\) and \(\alpha\) are constants
    • \(R > 0\)
    • \(0 < \alpha < \frac { \pi } { 2 }\)
    Give the exact value of \(R\) and the value of \(\alpha\) in radians to 3 decimal places. The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside a rabbit hole on a particular day is modelled by the equation $$\theta = 6.5 + 3 \cos \left( \frac { \pi t } { 13 } - 4 \right) + \sin \left( \frac { \pi t } { 13 } - 4 \right) \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight.
    Using the equation of the model and your answer to part (a)
    1. deduce the minimum value of \(\theta\) during this day,
    2. find the time of day when this minimum value occurs, giving your answer to the nearest minute.
  3. Find the rate of temperature increase in the rabbit hole at midday.
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [TURN OVER FOR QUESTION 7]
SPS SPS SM Statistics 2022 February Q7
7. The time, \(T\) seconds, that a pendulum takes to complete one swing is modelled by the formula $$T = a l ^ { b }$$ where \(l\) metres is the length of the pendulum and \(a\) and \(b\) are constants.
  1. Show that this relationship can be written in the form $$\log _ { 10 } T = b \log _ { 10 } l + \log _ { 10 } a$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{59120894-c480-492b-a304-106ddbadacf0-18_613_926_699_699} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A student carried out an experiment to find the values of the constants \(a\) and \(b\).
    The student recorded the value of \(T\) for different values of \(l\).
    Figure 3 shows the linear relationship between \(\log _ { 10 } l\) and \(\log _ { 10 } T\) for the student's data. The straight line passes through the points \(( - 0.7,0 )\) and \(( 0.21,0.45 )\) Using this information,
  2. find a complete equation for the model in the form $$T = a l ^ { b }$$ giving the value of \(a\) and the value of \(b\), each to 3 significant figures.
  3. With reference to the model, interpret the value of the constant \(a\).
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [TURN OVER FOR QUESTION 8]
SPS SPS SM Statistics 2022 February Q8
8.
  1. Use the substitution \(u = 1 + \sin ^ { 2 } x\) to show that $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 8 \tan x } { 1 + \sin ^ { 2 } x } \mathrm {~d} x = \int _ { p } ^ { q } \frac { 4 } { u ( 2 - u ) } \mathrm { d } u$$ where \(p\) and \(q\) are constants to be found.
  2. Hence, using algebraic integration, show that $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 8 \tan x } { 1 + \sin ^ { 2 } x } \mathrm {~d} x = \ln A$$ where \(A\) is a rational number to be found.
    [0pt] [BLANK PAGE]
SPS SPS SM Statistics 2022 February Q9
9. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
The first 3 terms of an arithmetic sequence are $$\ln 3 \quad \ln \left( 3 ^ { k } - 1 \right) \quad \ln \left( 3 ^ { k } + 5 \right)$$ Find the exact value of the constant \(k\).
[0pt] [BLANK PAGE]
SPS SPS SM Statistics 2022 February Q10
10. (i) Use proof by exhaustion to show that for \(n \in \mathbb { N } , n \leqslant 4\) $$( n + 1 ) ^ { 3 } > 3 ^ { n }$$ (ii) Given that \(m ^ { 3 } + 5\) is odd, use proof by contradiction to show, using algebra, that \(m\) is even.
[0pt] [BLANK PAGE]
SPS SPS SM Statistics 2022 February Q11
  1. Answer all the questions.
The discrete random variable \(X\) has the probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c c } c ( 7 - 2 x ) & x = 0,1,2,3
k & x = 4
0 & \text { otherwise } \end{array} \right.$$ where \(c\) and \(k\) are constants.
  1. Show that \(16 c + k = 1\)
  2. Given that \(\mathrm { P } ( X \geq 3 ) = \frac { 5 } { 8 }\) find the value of \(c\) and the value of \(k\).
    [0pt] [BLANK PAGE]
SPS SPS SM Statistics 2022 February Q12
12. The four pie charts illustrate the numbers of employees using different methods of travel in four Local Authorities in 2011.
\includegraphics[max width=\textwidth, alt={}, center]{59120894-c480-492b-a304-106ddbadacf0-30_1234_1160_276_255} \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Key:}
Public transport
\(\therefore\)Private motorised transport
\(\therefore\)Bicycle
All other methods of travel
\end{table}
  1. State, with reasons, which of the four Local Authorities is most likely to be a rural area with many hills.
  2. Two of the Local Authorities represent urban areas.
    1. State with a reason which two Local Authorities are likely to be urban.
    2. One urban Local Authority introduced a Park-and-Ride service in 2006. Users of this service drive to the edge of the urban area and then use buses to take them into the centre of the area. A student claims that a comparison of the corresponding pie charts for 2001 (not shown) and 2011 would enable them to identify which Local Authority this was. State with a reason whether you agree with the student.
      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]