Questions S1 (2020 questions)

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Edexcel S1 Q1
8 marks Easy -1.2
    1. Name a suitable distribution for modelling the volume of liquid in bottles of wine sold as containing 75 cl.
    2. Explain why the mean in such a model would probably be greater than 75 cl.
    [2 marks]
    1. Name a suitable distribution for modelling the score on a single throw of a fair four-sided die with the numbers 1, 2, 3 and 4 on its faces.
    2. Use your suggested model to find the mean and variance of the score on a single throw of the die.
    [6 marks]
Edexcel S1 Q2
10 marks Moderate -0.3
The events \(A\) and \(B\) are independent and such that $$\text{P}(A) = 2\text{P}(B) \text{ and } \text{P}(A \cap B) = \frac{1}{8}.$$
  1. Show that \(\text{P}(B) = \frac{1}{4}\). [5 marks]
  2. Find \(\text{P}(A \cup B)\). [3 marks]
  3. Find \(\text{P}(A | B')\). [2 marks]
Edexcel S1 Q3
11 marks Standard +0.3
A call-centre dealing with complaints collected data on how long customers had to wait before an operator was free to take their call. The lower quartile of the data was 12.7 minutes and the interquartile range was 5.8 minutes.
  1. Find the value of the upper quartile of the data. [1 mark]
It is suggested that a normal distribution could be used to model the waiting time.
  1. Calculate correct to 3 significant figures the mean and variance of this normal distribution based on the values of the quartiles. [8 marks]
The actual mean and variance of the data were 15.3 minutes and 20.1 minutes\(^2\) respectively.
  1. Comment on the suitability of the model. [2 marks]
Edexcel S1 Q4
14 marks Moderate -0.8
A College offers evening classes in GCSE Mathematics and English. In order to assess which age groups were reluctant to use the classes, the College collected data on the age in completed years of those currently attending each course. The results are shown in this back-to-back stem and leaf diagram. \includegraphics{figure_4} Key: \(1 | 3 | 2\) means age 31 doing Mathematics and age 32 doing English
  1. Find the median and quartiles of the age in completed years of those attending the Mathematics classes. [4 marks]
  2. On graph paper, draw a box plot representing the data for the Mathematics class. [3 marks]
The median and quartiles of the age in completed years of those attending the English classes are 25, 41 and 57 years respectively.
  1. Draw a box plot representing the data for the English class using the same scale as for the data from the Mathematics class. [3 marks]
  2. Using your box plots, compare and contrast the ages of those taking each class. [4 marks]
Edexcel S1 Q5
16 marks Moderate -0.3
A netball team are in a league with three other teams from which one team will progress to the next stage of the competition. The team's coach estimates their chances of winning each of their three matches in the league to be 0.6, 0.5 and 0.3 respectively, and believes these probabilities to be independent of each other.
  1. Show that the probability of the team winning exactly two of their three matches is 0.36 [4 marks]
Let the random variable \(W\) be the number of matches that the team win in the league.
  1. Find the probability distribution of \(W\). [4 marks]
  2. Find E\((W)\) and Var\((W)\). [6 marks]
  3. Comment on the coach's assumption that the probabilities of success in each of the three matches are independent. [2 marks]
Edexcel S1 Q6
16 marks Moderate -0.3
The Principal of a school believes that more students are absent on days when the temperature is lower. Over a two-week period in December she records the percentage of students who are absent, \(A\%\), and the temperature, \(T°\)C, at 9 am each morning giving these results.
\(T\) (°C)4\(-3\)\(-2\)\(-6\)037\(-1\)32
\(A\) (\%)8.514.117.020.317.915.512.412.813.711.6
  1. Represent these data on a scatter diagram. [4 marks]
You may use $$\Sigma T = 7, \quad \Sigma A = 143.8, \quad \Sigma T^2 = 137, \quad \Sigma A^2 = 2172.66, \quad \Sigma TA = 20.7$$
  1. Calculate the product moment correlation coefficient for these data and comment on the Principal's hypothesis. [6 marks]
  2. Find an equation of the regression line of \(A\) on \(T\) in the form \(A = p + qT\). [4 marks]
  3. Draw the regression line on your scatter diagram. [2 marks]
OCR MEI S1 Q1
8 marks Moderate -0.8
Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope. The probability distribution for the number of letters, \(X\), which are now in the correct envelope is given in the following table.
\(r\)01
P(X = r)\(\frac{3}{8}\)\(\frac{1}{3}\)\(\frac{1}{4}\)0\(\frac{1}{24}\)
  1. Explain why the case \(X = 3\) is impossible. [1]
  2. Explain why P(\(X = 4\)) = \(\frac{1}{24}\). [2]
  3. Calculate E(\(X\)) and Var(\(X\)). [5]
OCR MEI S1 Q2
5 marks Easy -1.2
A company sells sugar in bags which are labelled as containing 450 grams. Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.
  1. State two adjustments the company could make. [2]
The weights, \(x\) grams, of a random sample of 25 bags are now recorded.
  1. Given that \(\sum x = 11409\) and \(\sum x^2 = 5206937\), calculate the sample mean and sample standard deviation of these weights. [3]
OCR MEI S1 Q3
8 marks Moderate -0.8
Jeremy is a computing consultant who sometimes works at home. The number, \(X\), of days that Jeremy works at home in any given week is modelled by the probability distribution P(\(X = r\)) = \(\frac{1}{40}r(r + 1)\) for \(r = 1, 2, 3, 4\).
  1. Verify that P(\(X = 4\)) = \(\frac{1}{2}\). [1]
  2. Calculate E(\(X\)) and Var(\(X\)). [5]
  3. Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days. [2]
OCR MEI S1 Q4
7 marks Moderate -0.8
A sprinter runs many 100-metre trials, and the time, \(x\) seconds, for each is recorded. A sample of eight of these times is taken, as follows. 10.53 \quad 10.61 \quad 10.04 \quad 10.49 \quad 10.63 \quad 10.55 \quad 10.47 \quad 10.63
  1. Calculate the sample mean, \(\bar{x}\), and sample standard deviation, \(s\), of these times. [3]
  2. Show that the time of 10.04 seconds may be regarded as an outlier. [2]
  3. Discuss briefly whether or not the time of 10.04 seconds should be discarded. [2]
OCR MEI S1 Q5
6 marks Moderate -0.8
The number, \(X\), of children per family in a certain city is modelled by the probability distribution P(\(X = r\)) = \(k(6 - r)(1 + r)\) for \(r = 0, 1, 2, 3, 4\).
  1. Copy and complete the following table and hence show that the value of \(k\) is \(\frac{1}{50}\). [3]
    \(r\)01234
    P(\(X = r\))\(6k\)\(10k\)
  2. Calculate E(\(X\)). [2]
  3. Hence write down the probability that a randomly selected family in this city has more than the mean number of children. [1]
OCR MEI S1 Q6
17 marks Moderate -0.8
The weights, \(w\) grams, of a random sample of 60 carrots of variety A are summarised in the table below.
Weight\(30 \leqslant w < 50\)\(50 \leqslant w < 60\)\(60 \leqslant w < 70\)\(70 \leqslant w < 80\)\(80 \leqslant w < 90\)
Frequency111018147
  1. Draw a histogram to illustrate these data. [5]
  2. Calculate estimates of the mean and standard deviation of \(w\). [4]
  3. Use your answers to part (ii) to investigate whether there are any outliers. [3]
The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. \(n = 50\) \quad \(\sum x = 3624.5\) \quad \(\sum x^2 = 265416\)
  1. Calculate the mean and standard deviation of \(x\). [3]
  2. Compare the central tendency and variation of the weights of varieties A and B. [2]
OCR MEI S1 Q7
7 marks Moderate -0.8
A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a £10 prize, 20 of them have a £100 prize, one of them has a £5000 prize and all of the rest have no prize. This information is summarised in the frequency table below.
Prize money£0£10£100£5000
Frequency992950201
  1. Find the mean and standard deviation of the prize money per ticket. [4]
  2. I buy two of these tickets at random. Find the probability that I win either two £10 prizes or two £100 prizes. [3]
OCR MEI S1 Q1
8 marks Moderate -0.8
It is known that 8% of the population of a large city use a particular web browser. A researcher wishes to interview some people from the city who use this browser. He selects people at random, one at a time.
  1. Find the probability that the first person that he finds who uses this browser is
    1. the third person selected, [3]
    2. the second or third person selected. [2]
  2. Find the probability that at least one of the first 20 people selected uses this browser. [3]
OCR MEI S1 Q2
8 marks Standard +0.3
Jimmy and Alan are playing a tennis match against each other. The winner of the match is the first player to win three sets. Jimmy won the first set and Alan won the second set. For each of the remaining sets, the probability that Jimmy wins a set is • 0.7 if he won the previous set, • 0.4 if Alan won the previous set. It is not possible to draw a set.
  1. Draw a probability tree diagram to illustrate the possible outcomes for each of the remaining sets. [3]
  2. Find the probability that Alan wins the match. [3]
  3. Find the probability that the match ends after exactly four sets have been played. [2]
OCR MEI S1 Q3
6 marks Moderate -0.8
In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random. • \(T\) is the event that this person likes tomato soup. • \(M\) is the event that this person likes mushroom soup. You are given that \(\text{P}(T) = 0.55\), \(\text{P}(M) = 0.33\) and \(\text{P}(T|M) = 0.80\).
  1. Use this information to show that the events \(T\) and \(M\) are not independent. [1]
  2. Find \(\text{P}(T \cap M)\). [2]
  3. Draw a Venn diagram showing the events \(T\) and \(M\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
OCR MEI S1 Q4
4 marks Easy -1.2
25% of the plants of a particular species have red flowers. A random sample of 6 plants is selected.
  1. Find the probability that there are no plants with red flowers in the sample. [2]
  2. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers. [2]
OCR MEI S1 Q5
8 marks Moderate -0.8
In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that \(\text{P}(W) = 0.14\), \(\text{P}(F) = 0.41\) and \(\text{P}(W \cap F) = 0.11\).
  1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
  2. Determine whether the events \(W\) and \(F\) are independent. [2]
  3. Find \(\text{P}(W|F)\) and explain what this probability represents. [3]
OCR MEI S1 Q6
4 marks Moderate -0.8
The table shows all the possible products of the scores on two fair four-sided dice. \includegraphics{figure_6}
  1. Find the probability that the product of the two scores is less than 10. [1]
  2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent. [3]
OCR MEI S1 Q7
8 marks Moderate -0.8
Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_7} A day is selected at random. Find the probability that
  1. the weather is wet and Andy travels by bus, [2]
  2. Andy walks or travels by bike, [3]
  3. the weather is dry given that Andy walks or travels by bike. [3]