Questions — SPS SPS FM Statistics (95 questions)

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SPS SPS FM Statistics 2023 January Q1
1. Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.
  1. Using a suitable model, find the probability of more than 4 calls arriving in a particular 20-minute period one morning. Indre is allowed 20 minutes of break time during each 4 -hour morning shift, which she can take in 5-minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed. One morning Indre took her break time in 4 periods of 5 minutes each.
  2. Find the probability that in exactly 3 of these periods there were no calls.
SPS SPS FM Statistics 2023 January Q2
2. A machine is set to fill pots with yoghurt such that the mean weight of yoghurt in a pot is 505 grams. To check that the machine is working properly, a random sample of 8 pots is selected. The weight of yoghurt, in grams, in each pot is as follows $$\begin{array} { l l l l l l l l } 508 & 510 & 500 & 500 & 498 & 503 & 508 & 505 \end{array}$$ Given that the weights of the yoghurt delivered by the machine follow a normal distribution with standard deviation 5.4 grams,
  1. find a \(95 \%\) confidence interval for the mean weight, \(\mu\) grams, of yoghurt in a pot. Give your answers to 2 decimal places.
SPS SPS FM Statistics 2023 January Q3
3. A large field of wheat is split into 8 plots of equal area. Each plot is treated with a different amount of fertiliser, \(f\) grams \(/ \mathrm { m } ^ { 2 }\). The yield of wheat, \(w\) tonnes, from each plot is recorded. The results are summarised below. $$\sum f = 28 \quad \sum w = 303 \quad \sum w ^ { 2 } = 13447 \quad \mathrm {~S} _ { f f } = 42 \quad \mathrm {~S} _ { f w } = 269.5$$
  1. Calculate the product moment correlation coefficient between \(f\) and \(w\)
  2. Interpret the value of your product moment correlation coefficient.
  3. Find the equation of the regression line of \(w\) on \(f\) in the form \(w = a + b f\)
SPS SPS FM Statistics 2023 January Q4
4. Sweet pea plants grown using a standard plant food have a mean height of 1.6 m . A new plant food is used for a random sample of 49 randomly chosen plants and the heights, \(x\) metres, of this sample can be summarised by the following. $$\begin{aligned} n & = 49
\Sigma x & = 74.48
\Sigma x ^ { 2 } & = 120.8896 \end{aligned}$$
  1. Test, at the \(5 \%\) significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m .
  2. State with a reason whether you needed to use the Central Limit Theorem to carry out the test in part (a).
SPS SPS FM Statistics 2023 January Q5
5. Nine athletes, \(A , B , C , D , E , F , G , H\) and \(I\), competed in both the 100 m sprint and the long jump. After the two events the positions of each athlete were recorded and Spearman's rank correlation coefficient was calculated and found to be 0.85
  1. Stating your hypotheses clearly, test whether or not there is evidence to suggest that the higher an athlete's position is in the 100 m sprint, the higher their position is in the long jump. Use a \(5 \%\) level of significance. The piece of paper the positions were recorded on was mislaid. Although some of the athletes agreed their positions, there was some disagreement between athletes \(B , C\) and \(D\) over their long jump results. The table shows the results that are agreed to be correct.
    Athlete\(A\)B\(C\)D\(E\)\(F\)G\(H\)I
    Position in \(\mathbf { 1 0 0 ~ m }\) sprint467928315
    Position in long jump549312
    Given that there were no tied ranks,
  2. find the correct positions of athletes \(B , C\) and \(D\) in the long jump. You must show your working clearly and give reasons for your answers.
    (5)
  3. Without recalculating the coefficient, explain how Spearman's rank correlation coefficient would change if athlete \(H\) was disqualified from both the 100 m sprint and the long jump.
    (2)
SPS SPS FM Statistics 2023 January Q6
6. A manufacturer makes two versions of a toy. One version is made out of wood and the other is made out of plastic. The weights, \(W \mathrm {~kg}\), of the wooden toys are normally distributed with mean 2.5 kg and standard deviation 0.7 kg . The weights, \(X \mathrm {~kg}\), of the plastic toys are normally distributed with mean 1.27 kg and standard deviation 0.4 kg . The random variables \(W\) and \(X\) are independent.
  1. Find the probability that the weight of a randomly chosen wooden toy is more than double the weight of a randomly chosen plastic toy.
    (6) The manufacturer packs \(n\) of these wooden toys and \(2 n\) of these plastic toys into the same container. The maximum weight the container can hold is 252 kg . The probability of the contents of this container being overweight is 0.2119 to 4 decimal places.
  2. Calculate the value of \(n\). END OF TEST
SPS SPS FM Statistics 2024 January Q1
1. The continuous random variable \(X\) has the distribution \(\mathrm { N } ( \mu , 30 )\). The mean of a random sample of 8 observations of \(X\) is 53.1. Determine a \(95 \%\) confidence interval for \(\mu\). You should give the end points of the interval correct to 4 significant figures.
SPS SPS FM Statistics 2024 January Q2
2. 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.
SPS SPS FM Statistics 2024 January Q3
3. Eight runners took part in two races. The positions in which the runners finished in the two races are shown in the table.
RunnerABCDEFGH
First race31562874
Second race43872561
Test at the \(5 \%\) significance level whether those runners who do better in one race tend to do better in the other.
SPS SPS FM Statistics 2024 January Q4
4. The manager of a car breakdown service uses the distribution \(\operatorname { Po } ( 2.7 )\) to model the number of punctures, \(R\), in a 24-hour period in a given rural area. The manager knows that, for this model to be valid, punctures must occur randomly and independently of one another.
  1. State a further assumption needed for the Poisson model to be valid.
  2. State the value of the standard deviation of \(R\).
  3. Use the model to calculate the probability that, in a randomly chosen period of 168 hours, at least 22 punctures occur. The manager uses the distribution \(\mathrm { Po } ( 0.8 )\) to model the number of flat batteries in a 24 -hour period in the same rural area, and he assumes that instances of flat batteries are independent of punctures. A day begins and ends at midnight, and a "bad" day is a day on which there are more than 6 instances, in total, of punctures and flat batteries.
  4. Assume first that both the manager's models are correct. Calculate the probability that a randomly chosen day is a "bad" day.
  5. It is found that 12 of the next 100 days are "bad" days. Comment on whether this casts doubt on the validity of the manager's models.
SPS SPS FM Statistics 2024 January Q5
5. A company uses two drivers for deliveries.
Driver \(A\) charges a fixed rate of \(\pounds 80\) per day plus \(\pounds 2\) per mile travelled on that day.
Driver \(B\) charges a fixed rate of \(\pounds 120\) per day plus \(\pounds 1.50\) per mile travelled on that day.
On each working day the total distance, in miles, travelled by each driver is a random variable with the distribution \(\mathrm { N } ( 83,360 )\). Find the probability that the total charge to the company of three randomly chosen days' deliveries by driver \(A\) is at least \(\pounds 300\) more than the total charge of two randomly chosen days' deliveries by driver \(B\).
SPS SPS FM Statistics 2024 January Q6
6. A firm claims that no more than \(2 \%\) of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than \(2 \%\). In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the \(5 \%\) significance level, using the null hypothesis \(p = 0.02\).
  1. Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect.
  2. The researcher finds that 18 out of the 600 packets are underweight. A colleague says
    " 18 out of 600 is \(3 \%\), so there is evidence that the actual proportion of underweight bags is greater than \(2 \%\)." Criticise this statement.
SPS SPS FM Statistics 2024 January Q7
7. The random variable \(X\) was assumed to have a normal distribution with mean \(\mu\). Using a random sample of size 128, a significance test was carried out using the following hypotheses.
\(\mathrm { H } _ { 0 } : \mu = 30\)
\(\mathrm { H } _ { 1 } : \mu > 30\)
It was found that \(\sum x = 3929.6\) and \(\sum x ^ { 2 } = 123483.52\). The conclusion of the test was to reject the null hypothesis.
  1. Determine the range of possible values of the significance level of the test.
  2. It was subsequently found that \(X\) was not normally distributed. Explain whether this invalidates the conclusion of the test.
SPS SPS FM Statistics 2024 January Q8
8. A teacher has 10 different mathematics books. Of these books, 5 are on Algebra, 3 are on Calculus and 2 are on Trigonometry. The teacher arranges all 10 books in random order on a shelf.
a) Find the probability that the Calculus books are next to each other and the Trigonometry books are next to each other. \section*{In this question you must show detailed reasoning.} b) Find the probability that 2 of the Calculus books are next to each other but the third Calculus book is separated from the other 2 by at least 1 other book.
SPS SPS FM Statistics 2024 January Q9
9. The continuous random variable \(X\) has a uniform distribution on the interval \([ - \pi , \pi ]\).
The random variable \(Y\) is defined by \(Y = \sin X\). Determine the cumulative distribution function of \(Y\). END OF TEST
SPS SPS FM Statistics 2025 April Q2
6 marks
2. 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:
    i) the median
    ii) the mean
    ii) the standard deviation
    [0pt] [4]
  6. Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context
    [0pt] [2]
SPS SPS FM Statistics 2025 April Q3
2 marks
3. 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 \(P ( M = 6 ) = \frac { 1 } { 3 }\)
    [0pt] [2] 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 \(P ( S = 7 \mid M = S )\)
SPS SPS FM Statistics 2025 April Q4
4. The discrete random variable \(X\) has a geometric distribution. It is given that \(\operatorname { Var } ( X ) = 20\). Determine \(\mathrm { P } ( X \geqslant 7 )\).
SPS SPS FM Statistics 2025 April Q5
7 marks
5. An examination paper consists of 8 questions, of which one is on geometric distributions and one is on binomial distributions.
  1. If the 8 questions are arranged in a random order, find the probability that the question on geometric distributions is next to the question on binomial distributions.
    [0pt] [2]
    Four of the questions, including the one on geometric distributions, are worth 7 marks each, and the remaining four questions, including the one on binomial distributions, are worth 9 marks each. The 7 -mark questions are the first four questions on the paper, but are arranged in random order. The 9 -mark questions are the last four questions, but are arranged in random order. Find the probability that
  2. the questions on geometric distributions and on binomial distributions are next to one another,
    [0pt] [2]
  3. the questions on geometric distributions and on binomial distributions are separated by at least 2 other questions.
    [0pt] [3] \section*{6.} The random variable \(X\) represents the weight in kg of a randomly selected male dog of a particular breed. \(X\) is Normally distributed with mean 30.7 and standard deviation 3.5.
    i) Find the \(90 ^ { \text {th } }\) percentile for the weights of these dogs.
    ii) Five of these dogs are chosen at random. Find the probability that exactly four of them weighs at least 30 kg . The weights of females of the same breed of dog are Normally distributed with mean 26.8 kg .
    iii) Given that \(5 \%\) of female dogs of this breed weigh more than 30 kg , find the standard deviation of their weights.
    iv) Sketch the distributions of the weights of male and female dogs of this breed on a single diagram.
SPS SPS FM Statistics 2025 April Q7
7. The random variable \(y\) has probability density function \(\mathrm { f } ( y )\) given by $$\mathrm { f } ( y ) = \left\{ \begin{array} { c c } k y ( a - y ) & 0 \leq y \leq 3
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are positive constants.
    1. Explain why \(a \geq 3\)
    2. Show that \(k = \frac { 2 } { 9 ( a - 2 ) }\) Given that \(\mathrm { E } ( Y ) = 1.75\)
  2. Find the values of a and k .
  3. Write down the mode of Y
SPS SPS FM Statistics 2024 September Q1
1. The Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students.
\includegraphics[max width=\textwidth, alt={}, center]{a65400d1-fadc-4bc7-ba4b-af2df57e390a-04_551_894_395_169} 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.
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2024 September Q2
2.
  1. A team of 9 is chosen at random from a class consisting of 8 boys and 12 girls. Find the probability that the team contains no more than 3 girls.
  2. A group of \(n\) people, including Mr and Mrs Laplace, are arranged at random in a line. The probability that Mr and Mrs Laplace are placed next to each other is less than 0.1 . Find the smallest possible value of \(n\).
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2024 September Q3
3. The random variable \(D\) has the distribution \(\operatorname { Geo } ( p )\). It is given that \(\operatorname { Var } ( D ) = \frac { 40 } { 9 }\).
Determine
  1. \(\operatorname { Var } ( 3 D + 5 )\),
  2. \(\mathrm { E } ( 3 D + 5 )\),
  3. \(\mathrm { P } ( D > \mathrm { E } ( D ) )\).
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2024 September Q4
4. The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 22 ) = 0.242\). Find the value of \(\mu\).
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2024 September Q5
5. At a factory that makes crockery the quality control department has found that \(10 \%\) of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).
  1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. Assume now that your model is valid.
  2. Find
    (a) \(\mathrm { P } ( X = 3 )\),
  3. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3 .
    [0pt] [BLANK PAGE]