Questions — SPS SPS FM Statistics (26 questions)

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SPS SPS FM Statistics 2021 June Q1
4 marks Moderate -0.8
Employees at a company were asked how long their average commute to work was. The table below gives information about their answers.
Time taken (\(t\) minutes)Number of people
\(0 < t \leq 10\)\(x\)
\(10 < t \leq 20\)30
\(20 < t \leq 30\)35
\(30 < t \leq 50\)28
\(50 < t \leq 90\)12
The company estimates that the mean time for employees commuting to work is 28 minutes. Work out the value of \(x\), showing your working clearly. [4]
SPS SPS FM Statistics 2021 June Q2
8 marks Moderate -0.3
Events \(A\) and \(B\) are such that \(P(A \cup B) = 0.95\), \(P(A \cap B) = 0.6\) and \(P(A|B) = 0.75\).
  1. Find \(P(B)\). [3]
  2. Find \(P(A)\). [3]
  3. Show that the events \(A'\) and \(B\) are independent. [2]
SPS SPS FM Statistics 2021 June Q3
4 marks Standard +0.3
The letters of the word CHAFFINCH are written on cards.
  1. In how many ways can the letters be rearranged with no restrictions. [1]
  2. In how many difference ways can the letters be rearranged if the vowels are to have at least one consonant between them. [3]
SPS SPS FM Statistics 2021 June Q4
9 marks Standard +0.3
The weights of sacks of potatoes are normally distributed. It is known that one in five sacks weigh more than 6kg and three in five sacks weigh more than 5.5kg.
  1. Find the mean and standard deviation of the weights of potato sacks. [5]
  2. The sacks are put into crates, with twelve sacks going into each crate. What is the probability that a given crate contains two or more sacks that weigh more than 6kg? You must explain your reasoning clearly in this question. [4]
SPS SPS FM Statistics 2021 June Q5
7 marks Moderate -0.3
Eleven students in a class sit a Mathematics exam and their average score is 67% with a standard deviation of 12%. One student from the class is absent and sits the paper later, achieving a score of 85%.
  1. Find the mean score for the whole class and the standard deviation for the whole class. [5]
  2. Comment, with justification, on whether the score for the paper sat later should be considered as an outlier. [2]
SPS SPS FM Statistics 2021 June Q6
6 marks Standard +0.3
Only two airlines fly daily into an airport. AMP Air has 70 flights per day and Volt Air has 65 flights per day. Passengers flying with AMP Air have an 18% probability of losing their luggage and passengers flying with Volt Air have a 23% probability of losing their luggage. You overhear a passenger in the airport complaining about her luggage being lost. Find the exact probability that she travelled with Volt Air, giving your answer as a rational number. [6]
SPS SPS FM Statistics 2021 June Q7
12 marks Standard +0.8
A continuous random variable \(X\) has probability density function \(f\) given by $$f(x) = \begin{cases} \frac{x^2}{a} + b, & 0 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}$$ where \(a\) and \(b\) are positive constants. It is given that \(P(X \geq 2) = 0.75\).
  1. Show that \(a = 32\) and \(b = \frac{1}{12}\). [5]
  2. Find \(E(X)\). [3]
  3. Find \(P(X > E(X)|X > 2)\) [4]
SPS SPS FM Statistics 2021 January Q1
4 marks Moderate -0.3
Alan's journey time to work can be modelled by a normal distribution with standard deviation 6 minutes. Alan measures the journey time to work for a random sample of 5 journeys. The mean of the 5 journey times is 36 minutes.
  1. Construct a 95\% confidence interval for Alan's mean journey time to work, giving your values to one decimal place. [2 marks]
  2. Alan claims that his mean journey time to work is 30 minutes. State, with a reason, whether or not the confidence interval found in part (a) supports Alan's claim. [1 mark]
  3. Suppose that the standard deviation is not known but a sample standard deviation is found from Alan's sample and calculated to be 6. Explain how the working in part (a) would change. [1 mark]
SPS SPS FM Statistics 2021 January Q2
8 marks Standard +0.3
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. [3]
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.
  1. Find the probability that in exactly 3 of these periods there were no calls. [2]
On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.
  1. Find the probability that Indre missed exactly 1 call in each of these 2 breaks. [3]
SPS SPS FM Statistics 2021 January Q3
7 marks Moderate -0.3
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/m². The yield of wheat, \(w\) tonnes, from each plot is recorded. The results are summarised below. $$\sum f = 28 \quad \sum w = 303 \quad \sum w^2 = 13447 \quad S_{ff} = 42 \quad S_{fw} = 269.5$$
  1. Calculate the product moment correlation coefficient between \(f\) and \(w\) [2]
  2. Interpret the value of your product moment correlation coefficient. [1]
  3. Find the equation of the regression line of \(w\) on \(f\) in the form \(w = a + bf\) [3]
  4. Using your equation, estimate the decrease in yield when the amount of fertiliser decreases by 0.5 grams/m² [1]
SPS SPS FM Statistics 2021 January Q4
7 marks Standard +0.3
The continuous random variable \(X\) has cumulative distribution function given by $$F(x) = \begin{cases} 0 & x \leq 0 \\ k\left(x^3 - \frac{3}{8}x^4\right) & 0 < x \leq 2 \\ 1 & x > 2 \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac{1}{2}\) [1]
  2. Showing your working clearly, use calculus to find
    1. E(\(X\))
    2. the mode of \(X\)
    [6]
SPS SPS FM Statistics 2021 January Q5
9 marks Standard +0.3
A shopkeeper sells chocolate bars which are described by the manufacturer as having an average mass of 45 grams. The shopkeeper claims that the mass of the chocolate bars, \(X\) grams, is getting smaller on average. A random sample of 6 chocolate bars is taken and their masses in grams are measured. The results are $$\sum x = 246 \quad \text{and} \quad \sum x^2 = 10198$$ Investigate the shopkeeper's claim using the 5\% level of significance. State any assumptions that you make. [9 marks]
SPS SPS FM Statistics 2021 January Q6
12 marks Standard +0.3
A spinner can land on red or blue. When the spinner is spun, there is a probability of \(\frac{1}{3}\) that it lands on blue. The spinner is spun repeatedly. The random variable \(B\) represents the number of the spin when the spinner first lands on blue.
  1. Find
    1. P(\(B = 4\))
    2. P(\(B \leq 5\))
    [4]
  2. Find E(\(B^2\)) [3]
Steve invites Tamara to play a game with this spinner. Tamara must choose a colour, either red or blue. Steve will spin the spinner repeatedly until the spinner first lands on the colour Tamara has chosen. The random variable \(X\) represents the number of the spin when this occurs. If Tamara chooses red, her score is \(e^X\) If Tamara chooses blue, her score is \(X^2\)
  1. State, giving your reasons and showing any calculations you have made, which colour you would recommend that Tamara chooses. [5]
SPS SPS FM Statistics 2021 January Q7
7 marks Challenging +1.2
Nine athletes, \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\) and \(I\), competed in both the 100m 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 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 100m sprint467928315
Position in long jump549312
Given that there were no tied ranks,
  1. 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]
  2. Without recalculating the coefficient, explain how Spearman's rank correlation coefficient would change if athlete \(H\) was disqualified from both the 100m sprint and the long jump. [2]
SPS SPS FM Statistics 2021 September Q1
6 marks Moderate -0.8
  1. 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]
  2. 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]
SPS SPS FM Statistics 2021 September Q2
9 marks Moderate -0.3
\(P(E) = 0.25\), \(P(F) = 0.4\) and \(P(E \cap F) = 0.12\)
  1. Find \(P(E'|F')\) [2 marks]
  2. Explain, showing your working, whether or not \(E\) and \(F\) are statistically independent. Give reasons for your answer. [2 marks]
The event \(G\) has \(P(G) = 0.15\) The events \(E\) and \(G\) are mutually exclusive and the events \(F\) and \(G\) are independent.
  1. Draw a Venn diagram to illustrate the events \(E\), \(F\) and \(G\), giving the probabilities for each region. [3 marks]
  2. Find \(P([F \cup G]')\) [2 marks]
SPS SPS FM Statistics 2021 September Q3
11 marks Moderate -0.8
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.
  1. Draw a tree diagram to represent this information. [3 marks]
  2. Find the probability that a randomly selected student:
    1. always start their assignments on the day they are issued and hand them in on time. [2 marks]
    2. does not always hand in assignments on time and does not start their assignments on the day they are issued. [4 marks]
  3. 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. [2 marks]
SPS SPS FM Statistics 2021 September Q4
4 marks Moderate -0.8
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. [4 marks]
SPS SPS FM Statistics 2021 September Q5
9 marks Standard +0.3
The heights of a population of men are normally distributed with mean \(\mu\) cm and standard deviation \(\sigma\) cm. It is known that 20% of the men are taller than 180 cm and 5% are shorter than 170 cm.
  1. Sketch a diagram to show the distribution of heights represented by this information. [2 marks]
  2. Find the value of \(\mu\) and \(\sigma\). [5 marks]
  3. Three men are selected at random, find the probability that they are all taller than 175 cm. [2 marks]
SPS SPS FM Statistics 2025 April Q1
8 marks Moderate -0.3
It is known that, under standard conditions, 12% of light bulbs from a certain manufacturer have a defect. A quality improvement process has been implemented, and a random sample of 200 light bulbs produced after the improvements was selected. It was found that 15 of the 200 light bulbs were defective.
  1. State one assumption needed in order to use a binomial model for the number of defective light bulbs in the sample. [1]
  2. Test, at the 5% significance level, whether the proportion of defective light bulbs has decreased under the new process. [7]
SPS SPS FM Statistics 2025 April Q2
13 marks Moderate -0.8
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 | 3 5 7 9 21 | 0 2 5 6 8 22 | 1 3 4 5 7 9 23 | 0 2 5 8 24 | 1 4 6 7 25 | 2 5 Key: 21 | 0 represents a reaction time of 210 milliseconds
  1. State the median reaction time. [1]
  2. Calculate the interquartile range of these reaction times. [2]
  3. Find the mean and standard deviation of these reaction times. [3]
  4. State one advantage of using a stem-and-leaf diagram to display this data rather than a frequency table. [1]
  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:
    1. the median
    2. the mean
    3. the standard deviation
    [4]
  6. Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context [2]
SPS SPS FM Statistics 2025 April Q3
9 marks Standard +0.8
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 \(P(M = 6) = \frac{1}{3}\) [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.
  1. Find \(P(M = S)\) [3]
  2. Find \(P(S = 7 | M = S)\) [4]
SPS SPS FM Statistics 2025 April Q4
6 marks Standard +0.8
The discrete random variable \(X\) has a geometric distribution. It is given that \(\text{Var}(X) = 20\). Determine \(P(X \geq 7)\). [6]
SPS SPS FM Statistics 2025 April Q5
7 marks Standard +0.3
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. [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
  1. the questions on geometric distributions and on binomial distributions are next to one another, [2]
  2. the questions on geometric distributions and on binomial distributions are separated by at least 2 other questions. [3]
SPS SPS FM Statistics 2025 April Q6
11 marks Standard +0.3
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.
  1. Find the 90th percentile for the weights of these dogs. [2]
  2. Five of these dogs are chosen at random. Find the probability that exactly four of them weighs at least 30 kg. [3]
The weights of females of the same breed of dog are Normally distributed with mean 26.8 kg.
  1. Given that 5% of female dogs of this breed weigh more than 30 kg, find the standard deviation of their weights. [3]
  2. Sketch the distributions of the weights of male and female dogs of this breed on a single diagram. [3]