Questions — SPS SPS SM Statistics (77 questions)

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SPS SPS SM Statistics 2025 April Q4
4. A manufacturing plant produces electronic circuit boards that need to pass two quality checks - a mechanical inspection and an electrical test. Historical data shows that \(15 \%\) of boards fail the mechanical inspection. Of those that pass the mechanical inspection, \(8 \%\) fail the electrical test. Of those that fail the mechanical inspection, \(60 \%\) fail the electrical test.
  1. If a board is randomly selected from production, what is the probability that it passes both inspections?
  2. If a board is selected at random and is found to have passed the electrical test, what is the probability that it also passed the mechanical inspection?
  3. The company continues to test boards from a large batch until finding one that passes both inspections. Each board is tested independently of all others. What is the probability that they need to test exactly 3 boards to find one that passes both inspections?
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SPS SPS SM Statistics 2025 April Q5
4 marks
5. In a study of reaction times, 25 participants completed a test where their reaction times (in milliseconds) were recorded. The results are shown in the stem-and-leaf diagram below: \(20 \mid 3579\)
\(21 \mid 02568\)
\(22 \mid 134579\)
\(23 \mid 0258\)
\(24 \mid 1467\)
\(25 \mid 25\) Key: 21 | 0 represents a reaction time of 210 milliseconds
  1. State the median reaction time.
  2. Calculate the interquartile range of these reaction times.
  3. Find the mean and standard deviation of these reaction times.
  4. State one advantage of using a stem-and-leaf diagram to display this data rather than a frequency table.
  5. One participant completed the test again and recorded a reaction time of 195 milliseconds. Add this result to the stem-and-leaf diagram and state the effect this would have on:
    a. the median
    b. the mean
    c. the standard deviation
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  6. Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context
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SPS SPS SM Statistics 2025 April Q6
6. A retail bakery makes cherry muffins where, due to the production process, \(15 \%\) of muffins contain a lower than expected quantity of cherries. The bakery sells these muffins in boxes of 20.
  1. State a suitable distribution to model the number of muffins with a lower than expected quantity of cherries in a box, giving the value(s) of any parameter(s). State any assumptions needed for your model to be valid.
  2. Using your model from part (a), find the probability that a randomly selected box contains:
    1. exactly 3 muffins with a lower than expected quantity of cherries,
    2. at least 5 muffins with a lower than expected quantity of cherries.
  3. The bakery sells 25 boxes of muffins in one day. Find the probability that fewer than 4 of these boxes contain exactly 3 muffins with a lower than expected quantity of cherries.
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SPS SPS SM Statistics 2025 April Q7
7. Miguel has six numbered tiles, labelled \(2,2,3,3,4,4\). He selects two tiles at random, without replacement. The variable \(M\) denotes the sum of the numbers on the two tiles.
  1. Show that \(P ( M = 6 ) = \frac { 1 } { 3 }\) The table shows the probability distribution of \(M\)
    \(m\)45678
    \(P ( M = m )\)\(\frac { 1 } { 15 }\)\(\frac { 4 } { 15 }\)\(\frac { 1 } { 3 }\)\(\frac { 4 } { 15 }\)\(\frac { 1 } { 15 }\)
    Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable \(S\) denotes the sum of the numbers on the two tiles that Sofia selects.
  2. Find \(P ( M = S )\)
  3. Find \(P ( S = 7 \mid M = S )\)
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SPS SPS SM Statistics 2024 September Q1
1. The histogram shows information about the lengths, \(l\) centimetres, of a sample of worms of a certain species.
\includegraphics[max width=\textwidth, alt={}, center]{c5ea8584-939f-4627-8f81-bac60233d9a3-04_913_1303_392_175} The number of worms in the sample with lengths in the class \(3 \leqslant l < 4\) is 30 .
  1. Find the number of worms in the sample with lengths in the class \(0 \leqslant l < 2\).
  2. Find an estimate of the number of worms in the sample with lengths in the range \(4.5 \leqslant l < 5.5\).
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SPS SPS SM Statistics 2024 September Q2
2. A factory buys \(10 \%\) of its components from supplier \(A , 30 \%\) from supplier \(B\) and the rest from supplier \(C\). It is known that \(6 \%\) of the components it buys are faulty. Of the components bought from supplier \(A , 9 \%\) are faulty and of the components bought from supplier \(B , 3 \%\) are faulty.
  1. Find the percentage of components bought from supplier \(C\) that are faulty. A component is selected at random.
  2. Explain why the event "the component was bought from supplier \(B\) " is not statistically independent from the event "the component is faulty".
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SPS SPS SM Statistics 2024 September Q3
3. The discrete random variable \(X\) takes values \(1,2,3,4\) and 5 , and its probability distribution is defined as follows. $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 1
\frac { 1 } { 2 } \mathrm { P } ( X = x - 1 ) & x = 2,3,4,5
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that \(a = \frac { 16 } { 31 }\). The discrete probability distribution for \(X\) is given in the table.
    \(x\)12345
    \(\mathrm { P } ( X = x )\)\(\frac { 16 } { 31 }\)\(\frac { 8 } { 31 }\)\(\frac { 4 } { 31 }\)\(\frac { 2 } { 31 }\)\(\frac { 1 } { 31 }\)
  2. Find the probability that \(X\) is odd. Two independent values of \(X\) are chosen, and their sum \(S\) is found.
  3. Find the probability that \(S\) is odd.
  4. Find the probability that \(S\) is greater than 8 , given that \(S\) is odd. Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable \(Y\) defined as follows. $$\mathrm { P } ( Y = y + 1 ) = \frac { 1 } { 2 } \mathrm { P } ( Y = y ) \quad \text { for all positive integers } y .$$
  5. Find \(\mathrm { P } ( Y = 1 )\).
  6. Give a reason why one of the variables, \(X\) or \(Y\), might be more appropriate as a model for the number of attempts that Sheila needs to start her car.
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SPS SPS SM Statistics 2024 September Q4
4. The radar diagrams illustrate some population figures from the 2011 census results.
\includegraphics[max width=\textwidth, alt={}, center]{c5ea8584-939f-4627-8f81-bac60233d9a3-10_723_776_360_159}
\includegraphics[max width=\textwidth, alt={}, center]{c5ea8584-939f-4627-8f81-bac60233d9a3-10_725_775_358_1055} Each radius represents an age group, as follows:
Radius123456
Age
group
\(0 - 17\)\(18 - 29\)\(30 - 44\)\(45 - 59\)\(60 - 74\)\(75 +\)
The distance of each dot from the centre represents the number of people in the relevant age group.
  1. The scales on the two diagrams are different. State an advantage and a disadvantage of using different scales in order to make comparisons between the ages of people in these two Local Authorities.
  2. Approximately how many people aged 45 to 59 were there in Liverpool?
  3. State the main two differences between the age profiles of the two Local Authorities.
  4. James makes the following claim.
    "Assuming that there are no significant movements of population either into or out of the two regions, the 2021 census results are likely to show an increase in the number of children in Liverpool and a decrease in the number of children in Rutland." Use the radar diagrams to give a justification for this claim.
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SPS SPS SM Statistics 2024 September Q6
6. A television company believes that the proportion of households that can receive Channel C is 0.35 .
  1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the \(2.5 \%\) significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35.
  2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working.
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SPS SPS SM Statistics 2024 September Q7
7. The Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students.
\includegraphics[max width=\textwidth, alt={}, center]{c5ea8584-939f-4627-8f81-bac60233d9a3-16_542_883_459_148} A student is chosen at random from the 100 students. Then another student is chosen from the remaining students. Find the probability that the first student studies History and the second student studies Geography but not Psychology.
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SPS SPS SM Statistics 2025 January Q1
  1. Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.
  • \(A\) is the event that Isobel's parents watch a match.
  • \(B\) is the event that Isobel scores in a match.
You are given that \(\mathrm { P } ( B \mid A ) = \frac { 3 } { 7 }\) and \(\mathrm { P } ( A ) = \frac { 7 } { 10 }\).
  1. Calculate \(\mathrm { P } ( A \cap B )\). The probability that Isobel does not score and her parents do not attend is 0.1 .
  2. Draw a Venn diagram showing the events \(A\) and \(B\), and mark in the probability corresponding to each of the regions of your diagram.
  3. Are events \(A\) and \(B\) independent? Give a reason for your answer.
  4. By comparing \(\mathrm { P } ( B \backslash A )\) with \(\mathrm { P } ( B )\), explain why Isobel should ask her parents not to attend.
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SPS SPS SM Statistics 2025 January Q2
2. On average, \(25 \%\) of the packets of a certain kind of soup contain a voucher. Kim buys one packet of soup each week for 12 weeks. The number of vouchers she obtains is denoted by \(X\).
  1. State two conditions needed for \(X\) to be modelled by the distribution \(\mathrm { B } ( 12,0.25 )\). In the rest of this question you should assume that these conditions are satisfied.
  2. Find \(\mathrm { P } ( X \leqslant 6 )\). In order to claim a free gift, 7 vouchers are needed.
  3. Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.
  4. Find the probability that Kim will be able to claim a free gift in the 12th week but not before.
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SPS SPS SM Statistics 2025 January Q3
3. The continuous random variable \(T\) has mean \(\mu\) and standard deviation \(\sigma\). It is known that \(\mathrm { P } ( T < 140 ) = 0.01\) and \(\mathrm { P } ( T < 300 ) = 0.8\).
  1. Assuming that \(T\) is normally distributed, calculate the values of \(\mu\) and \(\sigma\). In fact, \(T\) represents the time, in minutes, taken by a randomly chosen runner in a public marathon, in which about \(10 \%\) of runners took longer than 400 minutes.
  2. State with a reason whether the mean of \(T\) would be higher than, equal to, or lower than the value calculated in part (i).
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SPS SPS SM Statistics 2025 January Q4
4. The table shows information about the time, \(t\) minutes correct to the nearest minute, taken by 50 people to complete a race.
Time (minutes)\(t \leqslant 27\)\(28 \leqslant t \leqslant 30\)\(31 \leqslant t \leqslant 35\)\(36 \leqslant t \leqslant 45\)\(46 \leqslant t \leqslant 60\)\(t \geqslant 61\)
Number of people04281440
  1. In a histogram illustrating the data, the height of the block for the \(31 \leqslant t \leqslant 35\) class is 5.6 cm . Find the height of the block for the \(28 \leqslant t \leqslant 30\) class. (There is no need to draw the histogram.)
  2. The data in the table are used to estimate the median time. State, with a reason, whether the estimated median time is more than 33 minutes, less than 33 minutes or equal to 33 minutes.
  3. Calculate estimates of the mean and standard deviation of the data.
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SPS SPS SM Statistics 2025 January Q5
5. The masses, \(m\) grams, of 52 apples of a certain variety were found and summarised as follows. $$n = 52 \quad \Sigma ( m - 150 ) = - 182 \quad \Sigma ( m - 150 ) ^ { 2 } = 1768$$ Calculate the variance and thus the exact value of \(\sum m ^ { 2 }\)
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SPS SPS SM Statistics 2025 January Q6
6. A television company believes that the proportion of households that can receive Channel C is 0.35 .
  1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the \(2.5 \%\) significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35 .
  2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working.
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SPS SPS SM Statistics 2025 January Q7
7. The table shows information, derived from the 2011 UK census, about the percentage of employees who used various methods of travel to work in four Local Authorities.
Local AuthorityUnderground, metro, light rail or tramTrainBusDriveWalk or cycle
A0.3\%4.5\%17\%52.8\%11\%
B0.2\%1.7\%1.7\%63.4\%11\%
C35.2\%3.0\%12\%11.7\%16\%
D8.9\%1.4\%9\%54.7\%10\%
One of the Local Authorities is a London borough and two are metropolitan boroughs, not in London.
  1. Which one of the Local Authorities is a London borough? Give a reason for your answer.
  2. Which two of the Local Authorities are metropolitan boroughs outside London? In each case give a reason for your answer.
  3. Describe one difference between the public transport available in the two metropolitan boroughs, as suggested by the table.
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SPS SPS SM Statistics 2025 January Q8
8. 80 randomly chosen people are asked to estimate a time interval of 60 seconds without using a watch or clock. The mean of the 80 estimates is 58.9 seconds. Previous evidence shows that the population standard deviation of such estimates is 5.0 seconds. Test, at the \(5 \%\) significance level, whether there is evidence that people tend to underestimate the time interval.
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SPS SPS SM Statistics 2026 January Q1
1. A telephone directory contains 50000 names. A researcher wishes to select a systematic sample of 100 names from the directory.
  1. Explain in detail how the researcher should obtain such a sample.
  2. Give one advantage and one disadvantage of
    1. quota sampling,
    2. systematic sampling.
SPS SPS SM Statistics 2026 January Q2
2. Each member of a group of 27 people was timed when completing a puzzle.
The time taken, \(x\) minutes, for each member of the group was recorded.
These times are summarised in the following box and whisker plot.
\includegraphics[max width=\textwidth, alt={}, center]{fdff6575-679e-4d25-ad43-e9d343c1746f-06_346_1383_427_278}
  1. Find the range of the times.
  2. Find the interquartile range of the times. For these 27 people \(\sum x = 607.5\) and \(\sum x ^ { 2 } = 17623.25\)
  3. calculate the mean time taken to complete the puzzle,
  4. calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.
  5. State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in \(a\) minutes and \(b\) minutes respectively, where \(a > b\).
    When their times are included with the data of the other 27 people
    • the median time increases
    • the mean time does not change
    • Suggest a possible value for \(a\) and a possible value for \(b\), explaining how your values satisfy the above conditions.
    • Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).
SPS SPS SM Statistics 2026 January Q3
3. Researchers investigated the change in the numbers of people in employment using underground, metro, light rail or tram (UMLRT) between 2001 and 2011. The data are combined for those Local Authorities (LAs) with UMLRT stations into five regions: Birmingham, Liverpool, Manchester, Sheffield and Rotherham, and Tyne and Wear. Fig. 1 shows the total numbers of people in employment in those LAs. Fig. 2 shows the total numbers of people in employment who use UMLRT in those LAs. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 1} \includegraphics[alt={},max width=\textwidth]{fdff6575-679e-4d25-ad43-e9d343c1746f-08_834_1694_836_166}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 2} \includegraphics[alt={},max width=\textwidth]{fdff6575-679e-4d25-ad43-e9d343c1746f-08_833_1694_1822_166}
\end{figure}
  1. Use these charts to explain which of Birmingham and Liverpool has the larger proportion of people in employment who used UMLRT in 2011. One of the researchers says, "Between 2001 and 2011, the increase in the number of people in employment who use UMLRT is greatest in Tyne and Wear." Sam says, "But what matters more is which region has the greatest increase in the proportion of people in employment who use UMLRT."
  2. Give a reason why the planners responsible for the building of trains and the maintenance of infrastructure might disagree with Sam.
  3. Explain whether those responsible for encouraging the greater use of public transport would agree with Sam.
  4. The charts are compiled from data in the Large Data Set by using those LAs which contain UMLRT stations in each region. Explain a disadvantage of using these data.
SPS SPS SM Statistics 2026 January Q4
4. Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard.
Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.
    1. Find the mean number of times he falls off in a day.
  1. (ii) Find the variance of the number of times he falls off in a day.
    1. Find the probability that, on a particular day, he falls off exactly 10 times.
  2. (ii) Find the probability that, on a particular day, he falls off 5 or more times.
  3. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
    1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days.
  4. (ii) Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days.
SPS SPS SM Statistics 2026 January Q5
5. The proportion of left-handed adults in a country is 10\%
Freya believes that the proportion of left-handed adults under the age of 25 in this country is different from 10\%
She takes a random sample of 40 adults under the age of 25 from this country to investigate her belief.
  1. Find the critical region for a suitable test to assess Freya's belief. You should
    • state your hypotheses clearly
    • use a \(5 \%\) level of significance
    • state the probability of rejection in each tail
    • Given the null hypothesis is true what is the probability of it being rejected in part (a)?
    In Freya's sample 7 adults were left-handed.
  2. With reference to your answer in part (a) comment on Freya's belief. \section*{6.}
SPS SPS SM Statistics 2026 January Q6
6. Skilled operators make a particular component for an engine. The company believes that the time taken to make this component may be modelled by the normal distribution. They timed one of their operators, Sheila, over a long period. They find that when she makes a component, she takes over 90 minutes to make one \(10 \%\) of the time, and that \(20 \%\) of the time, a component was less than 70 minutes to make. Estimate the mean and standard deviation of the time Sheila takes to make a component.
SPS SPS SM Statistics 2026 January Q7
7. A team game involves solving puzzles to escape from a room.
Using data from the past, the mean time to solve the puzzles and escape from one of these rooms is 65 minutes with a standard deviation of 11.3 minutes. After recent changes to the puzzles in the room, it is claimed that the mean time to solve the puzzles and escape has changed. To test this claim, a random sample of 100 teams is selected.
The total time to solve the puzzles and escape for the 100 teams is 6780 minutes.
Assuming that the times are normally distributed, test at the \(2 \%\) level the claim that the mean time has changed.