Questions — SPS SPS FM Statistics (95 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 FM Statistics 2021 September Q1
  1. a) 5 girls and 3 boys are arranged at random in a straight line. Find the probability that none of the boys is standing next to another boy.
    (3 marks)
    b) A cricket team consisting of six batsmen, four bowlers, and one wicket-keeper is to be selected from a group of 18 cricketers comprising nine batsmen, seven bowlers, and two wicket-keepers.
    How many different teams can be selected?
    (3 marks)
    [0pt] [BLANK PAGE]
  2. \(\quad \mathrm { P } ( E ) = 0.25 , \mathrm { P } ( F ) = 0.4\) and \(\mathrm { P } ( E \cap F ) = 0.12\)
    a Find \(P \left( E ^ { \prime } \mid F ^ { \prime } \right)\)
    b Explain, showing your working, whether or not \(E\) and \(F\) are statistically independent. Give reasons for your answer.
The event \(G\) has \(\mathrm { P } ( G ) = 0.15\)
The events \(E\) and \(G\) are mutually exclusive and the events \(F\) and \(G\) are independent.
c Draw a Venn diagram to illustrate the events \(E , F\) and \(G\), giving the probabilities for each region.
d Find \(\mathrm { P } \left( [ F \cup G ] ^ { \prime } \right)\)
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2021 September Q3
3. A group of students were surveyed by a principal and \(\frac { 2 } { 3 }\) were found to always hand in assignments on time. When questioned about their assignments \(\frac { 3 } { 5 }\) said they always start their assignments on the day they are issued and, of those who always start their assignments on the day they are issued, \(\frac { 11 } { 20 }\) hand them in on time.
a Draw a tree diagram to represent this information.
b Find the probability that a randomly selected student:
i always start their assignments on the day they are issued and hand them in on time.
ii does not always hand in assignments on time and does not start their assignments on the day they are issued.
c Determine whether or not always starting assignments on the day they are issued and handing them in on time are statistically independent. Give reasons for your answer.
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2021 September Q4
4. In a town, \(54 \%\) of the residents are female and \(46 \%\) are male. A random sample of 200 residents is chosen from the town. Using a suitable approximation, find the probability that more than half the sample are female.
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2021 September Q5
5. The heights of a population of men are normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm }\). It is known that \(20 \%\) of the men are taller than 180 cm and \(5 \%\) are shorter than 170 cm .
a Sketch a diagram to show the distribution of heights represented by this information.
b Find the value of \(\mu\) and \(\sigma\).
c Three men are selected at random, find the probability that they are all taller than 175 cm .
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q1
  1. The random variable \(X\) represents the clutch size (the number of eggs laid) by female birds of a particular species. The probability distribution of \(X\) is given in the table.
\(r\)234567
\(\mathrm { P } ( X = r )\)0.030.070.270.490.130.01
  1. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    On average \(65 \%\) of eggs laid result in a young bird successfully leaving the nest.
    1. Find the mean number of young birds that successfully leave the nest.
    2. Find the standard deviation of the number of young birds that successfully leave the nest.
      [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q2
2. A shopper estimates the cost, \(\pounds X\) per item, of each of 12 items in a supermarket. The shopper's estimates are compared with the actual cost, \(\pounds Y\) per item, of each item. The results are summarised as follows.
\(n = 12\)
\(\sum x ^ { 2 } = 28127\)
\(\sum x = 399\)
\(\Sigma y ^ { 2 } = 116509.0212\)
\(\Sigma y = 623.88\)
\(\sum x y = 45006.01\) Test at the \(1 \%\) significance level whether the shopper's estimates are positively correlated with the actual cost of the items.
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q3
3. A football player is practising taking penalties. On each attempt the player has a \(70 \%\) chance of scoring a goal. The random variable \(X\) represents the number of attempts that it takes for the player to score a goal.
  1. Determine \(\mathrm { P } ( X = 4 )\).
  2. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    • Determine the probability that the player needs exactly 4 attempts to score 2 goals.
      [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q4
  1. (a) Using the scatter diagram below, explain what is meant by least squares in the context of a regression line of \(y\) on \(x\).
    \includegraphics[max width=\textwidth, alt={}, center]{5a60e87d-7a09-4ef5-96ca-8f33030c8747-08_481_889_276_219}
    (b) A set of bivariate data \(( t , u )\) is summarised as follows.
$$\begin{array} { l l l } n = 5 & \sum t = 35 & \sum u = 54
\sum t ^ { 2 } = 285 & \sum u ^ { 2 } = 758 & \sum t u = 460 \end{array}$$
  1. Calculate the equation of the regression line of \(u\) on \(t\).
  2. The variables \(t\) and \(u\) are now scaled using the following scaling. $$v = 2 t , w = u + 4$$ Find the equation of the regression line of \(w\) on \(v\), giving your equation in the form $$w = \mathrm { f } ( v ) .$$ [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q5
5. Charlie carried out a survey on the main type of investment people have. The contingency table below shows the results of a survey of a random sample of people.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Main type of investment
\cline { 3 - 5 } \multicolumn{2}{c|}{}BondsCashStocks
\multirow{2}{*}{Age}\(25 - 44\)\(a\)\(b - e\)\(e\)
\cline { 2 - 5 }\(45 - 75\)\(c\)\(d - 59\)59
  1. Find an expression, in terms of \(a , b , c\) and \(d\), for the difference between the observed and the expected value \(( O - E )\) for the group whose main type of investment is Bonds and are aged \(45 - 75\)
    Express your answer as a single fraction in its simplest form. Given that \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.62\) for this information,
  2. test, at the \(5 \%\) level of significance, whether or not there is evidence of an association between the age of a person and the main type of investment they have. You should state your hypotheses, critical value and conclusion clearly. You may assume that no cells need to be combined.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q6
6. The 20 members of a club consist of 10 Town members and 10 Country members.
  1. All 20 members are arranged randomly in a straight line. Determine the probability that the Town members and the Country members alternate.
  2. Ten members of the club are chosen at random. Show that the probability that the number of Town members chosen is no more than \(r\), where \(0 \leqslant r \leqslant 10\), is given by
    \(\frac { 1 } { N } \sum _ { i = 0 } ^ { r } \left( { } ^ { 10 } C _ { i } \right) ^ { 2 }\)
    where \(N\) is an integer to be determined.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q7
7. (a) A substance emits particles randomly at a constant average rate of 3.2 per minute. A second substance emits particles randomly, and independently of the first source, at a constant average rate of 2.7 per minute. Find the probability that the total number of particles emitted by the two sources in a ten-minute period is less than 70 .
(b) The random variable \(X\) represents the number of particles emitted by a substance in a fixed time interval \(t\) minutes. It may be assumed that particles are emitted randomly and independently of each other. In general, the rate at which particles are emitted is proportional to the mass of the substance, but each particle emitted reduces the mass of the substance. Explain why a Poisson distribution may not be a valid model for \(X\) if the value of \(t\) is very large.
(c) The random variable \(Y\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that $$\begin{aligned} & \mathrm { P } ( Y = r ) = \mathrm { P } ( Y = r + 1 )
& \mathrm { P } ( Y = r ) = 1.5 \times \mathrm { P } ( Y = r - 1 ) \end{aligned}$$ Determine the following, in either order.
  • The value of \(r\)
  • The value of \(\lambda\)
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q1
  1. At a seaside resort the number \(X\) of ice-creams sold and the temperature \(Y ^ { \circ } \mathrm { F }\) were recorded on 20 randomly chosen summer days. The data can be summarised as follows.
$$\sum x = 1506 \quad \sum x ^ { 2 } = 127542 \quad \sum y = 1431 \quad \sum y ^ { 2 } = 104451 \quad \sum x y = 111297$$
  1. Calculate the equation of the least squares regression line of \(y\) on \(x\), giving your answer in the form \(y = a + b x\).
  2. Explain the significance for the regression line of the quantity \(\sum \left[ y _ { i } - \left( a x _ { i } + b \right) \right] ^ { 2 }\).
  3. It is decided to measure the temperature in degrees Centigrade instead of degrees Fahrenheit. If the same temperature is measured both as \(f ^ { \circ }\) Fahrenheit and \(c ^ { \circ }\) Centigrade, the relationship between \(f\) and \(c\) is \(c = \frac { 5 } { 9 } ( f - 32 )\). Find the equation of the new regression line.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q2
2. When babies are born, their head circumferences are measured. A random sample of 50 newborn female babies is selected. The sample mean head circumference is 34.711 cm . The sample standard deviation head circumference is 1.530 cm .
  1. Determine a \(95 \%\) confidence interval for the population mean head circumference of newborn female babies.
  2. Explain why you can calculate this interval even though the distribution of the population of head circumferences of newborn female babies is unknown.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q3
1 marks
3. In air traffic management, air traffic controllers send radio messages to pilots. On receiving a message, the pilot repeats it back to the controller to check that it has been understood correctly. At a particular site, on average \(4 \%\) of messages sent by controllers are not repeated back correctly and so have been misunderstood. You should assume that instances of messages being misunderstood occur randomly and independently.
  1. Find the probability that exactly 2 messages are misunderstood in a sequence of 50 messages.
  2. Find the probability that in a sequence of messages, the 10th message is the first one which is misunderstood.
    [0pt]
  3. Find the probability that in a sequence of 20 messages, there are no misunderstood messages. [1]
  4. Determine the expected number of messages required for 3 of them to be misunderstood.
  5. Determine the probability that in a sequence of messages, the 3rd misunderstood message is the 60th message in the sequence.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q4
4. Members of a photographic group may enter a maximum of 5 photographs into a members only competition.
Past experience has shown that the number of photographs, \(N\), entered by a member follows the probability distribution shown below.
\(n\)012345
\(\mathrm { P } ( N = n )\)\(a\)0.20.050.25\(b\)\(c\)
Given that \(\mathrm { E } ( 4 N + 2 ) = 14.8\) and \(\mathrm { P } ( N = 5 \mid N > 2 ) = \frac { 1 } { 2 }\)
  1. show that \(\operatorname { Var } ( N ) = 2.76\) The group decided to charge a 50 p entry fee for the first photograph entered and then 20 p for each extra photograph entered into the competition up to a maximum of \(\pounds 1\) per person. Thus a member who enters 3 photographs pays 90 p and a member who enters 4 or 5 photographs just pays £l Assuming that the probability distribution for the number of photographs entered by a member is unchanged,
  2. calculate the expected entry fee per member.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q5
3 marks
5. A practice examination paper is taken by 500 candidates, and the organiser wishes to know what continuous distribution could be used to model the actual time, \(X\) minutes, taken by candidates to complete the paper. The organiser starts by carrying out a goodness-of-fit test for the distribution \(\mathrm { N } \left( 100,15 ^ { 2 } \right)\) at the \(5 \%\) significance level. The grouped data and the results of some of the calculations are shown in the following table.
Time\(0 \leqslant X < 80\)\(80 \leqslant X < 90\)\(90 \leqslant X < 100\)\(100 \leqslant X < 110\)\(x \geqslant 110\)
Observed frequency \(O\)3695137129103
Expected frequency \(E\)45.60680.641123.754123.754126.246
\(\frac { ( O - E ) ^ { 2 } } { E }\)2.0232.5571.4180.2224.280
  1. State suitable hypotheses for the test.
  2. Show how the figures 123.754 and 0.222 in the column for \(100 \leqslant X < 110\) were obtained. [3]
  3. Carry out the test.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 February Q6
6. The continuous random variable \(Y\) has a uniform distribution on [ 0,2 ].
  1. It is given that \(\mathrm { E } [ a \cos ( a Y ) ] = 0.3\), where \(a\) is a constant between 0 and 1 , and \(a Y\) is measured in radians. Determine the value of the constant \(a\).
  2. Determine the \(60 ^ { \text {th } }\) percentile of \(Y ^ { 2 }\).
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2022 January Q1
  1. A local authority official wishes to conduct a survey of households in the borough. He decides to select a stratified sample of 2000 households using Council Tax property bands as the strata. At the time of the survey there are 79368 households in the borough. The table shows the numbers of households in the different tax bands.
Tax bandA-BC-DE-FG-H
Number of households322983321197394120
a. Calculate the number of households that the official should choose from each stratum in order to obtain his sample of 2000 households so that each stratum is represented proportionally.
b. State one advantage of stratified sampling over simple random sampling.
SPS SPS FM Statistics 2022 January Q2
2. A survey is carried out into the length of time for which customers wait for a response on a telephone helpline. A statistician who is analysing the results of the survey starts by modelling the waiting time, \(X\) minutes, by an exponential distribution with probability density function $$f ( x ) = \left\{ \begin{array} { c c } \lambda e ^ { - \lambda x } & x \geq 0
0 & x < 0 \end{array} \right.$$ The mean waiting time is found to be 5 minutes.
a. State the value of \(\lambda\).
b. Use the model to calculate the probability that a customer has to wait longer that 20 minutes for a response.
SPS SPS FM Statistics 2022 January Q3
3. A shop sells carrots and broccoli. The weights of carrots can be modelled by a normal distribution with mean 130 grams and variance 25 grams \(^ { 2 }\) and the weights of broccoli can be modelled by a normal distribution with mean 400 grams and variance 80 grams \({ } ^ { 2 }\). Find the probability that the weight of six randomly chosen carrots is more than two times the weight of one randomly chosen broccoli.
SPS SPS FM Statistics 2022 January Q4
4. The strength of beams compared against the moisture content of the beam is indicated in the following table.
Strength21.122.723.121.522.422.621.121.721.021.4
Moisture
content
11.18.98.88.98.89.910.710.510.510.7
a. Use your calculator to write down the value of the product moment correlation coefficient for these data.
b. Perform a two-tailed test, at the \(5 \%\) level of significance, to investigate whether there is correlation between strength and moisture content.
c. Use your calculator to write down the equation of the regression line of strength on moisture content.
d. Use the regression line to estimate the strength of a beam that has a moisture content of 9.5.
SPS SPS FM Statistics 2022 January Q5
5. In a large population of hens, the weight of a hen is normally distributed with mean \(\mu \mathrm { kg }\) and standard deviation \(\sigma \mathrm { kg }\). A random sample of 100 hens is taken from the population. The mean weight for the sample is denoted \(\bar { X }\).
a. State the distribution of \(\bar { X }\) giving its mean and variance. The sample values are summarised by \(\sum x = 199.8\) and \(\sum x ^ { 2 } = 407.8\) where \(x \mathrm {~kg}\) is the weight of a hen.
b. Find an unbiased estimate for \(\mu\).
c. Find an unbiased estimate for \(\sigma ^ { 2 }\).
d. Find a \(90 \%\) confidence interval for \(\mu\). It is found that \(\sigma = 0.27\). It is decided to test, at the \(1 \%\) level of significance, the null hypothesis \(\mu = 1.95\) against the alternative hypothesis \(\mu > 1.95\).
e. Find the \(p\)-value for the test.
f. Write down the conclusion reached.
g. Explain whether or not the central limit theorem was required in part e.
SPS SPS FM Statistics 2022 January Q6
6. The number of A-grades, \(X\), achieved in total by students at Lowkey School in their Mathematical examinations each year can be modelled by a Poisson distribution with a mean of 3 .
a. Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations.
b. The number of A-grades, \(Y\), achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with mean of 7 . Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A-grades in their Mathematics and English examinations.
c. Lowkey School is given a performance rating, \(P = 2 X + 3 Y\), based on the number of A-grades achieved in Mathematics and English. Find: $$\begin{array} { l l } \text { i. } & \mathrm { E } ( P )
\text { ii. } & \operatorname { Var } ( P ) \end{array}$$ d. What assumption did you make in answering part (b)? Did you need this assumption to answer part (c)? Justify your answers.
SPS SPS FM Statistics 2022 January Q7
7. The continuous random variable \(X\) has probability density function given by $$f ( x ) = \left\{ \begin{array} { c l } 0 & x < 1
\frac { 4 } { x ^ { 5 } } & x \geq 1 \end{array} \right.$$ a. Find the cumulative distribution function, \(F ( x )\), of \(X\).
b. Find the interquartile range of \(X\).
c. Show that the probability density function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), is given by $$g ( y ) = \left\{ \begin{array} { c l } 2 y & 0 < y \leq 1
0 & \text { otherwise } \end{array} \right.$$ d. Find the value of \(a\) for which \(\mathrm { E } \left( \frac { 1 } { X ^ { 2 } } \right) = a \mathrm { E } \left( X ^ { 2 } \right)\).
SPS SPS FM Statistics 2023 April Q1
21 marks
  1. \(\mathrm { E } ( a X + b Y + c ) = a \mathrm { E } ( X ) + b \mathrm { E } ( Y ) + c\),
  2. if \(X\) and \(Y\) are independent then \(\operatorname { Var } ( a X + b Y + c ) = a ^ { 2 } \operatorname { Var } ( X ) + b ^ { 2 } \operatorname { Var } ( Y )\).
\section*{Discrete distributions} \(X\) is a random variable taking values \(x _ { i }\) in a discrete distribution with \(\mathrm { P } \left( X = x _ { i } \right) = p _ { i }\)
Expectation: \(\mu = \mathrm { E } ( X ) = \sum x _ { i } p _ { i }\)
Variance: \(\sigma ^ { 2 } = \operatorname { Var } ( X ) = \sum \left( x _ { i } - \mu \right) ^ { 2 } p _ { i } = \sum x _ { i } ^ { 2 } p _ { i } - \mu ^ { 2 }\) Greg and Nilaya play a game with these dice.
Greg throws the black die and Nilaya throws the white die. Greg wins the game if he scores at least two more than Nilaya, otherwise Greg loses.
The probability of Greg winning the game is \(\frac { 1 } { 6 }\)
(b) Find the value of \(a\) and the value of \(b\) Show your working clearly. The random variable \(X = 2 W - 5\)
Given that \(\mathrm { E } ( X ) = 2.6\)
(c) find the exact value of \(\operatorname { Var } ( X )\) END OF EXAMINATION