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OCR MEI S1 2014 June 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 2014 June Q7
19 marks Standard +0.3
It is known that on average 85% of seeds of a particular variety of tomato will germinate. Ramesh selects 15 of these seeds at random and sows them.
    1. Find the probability that exactly 12 germinate. [3]
    2. Find the probability that fewer than 12 germinate. [2]
The following year Ramesh finds that he still has many seeds left. Because the seeds are now one year old, he suspects that the germination rate will be lower. He conducts a trial by randomly selecting \(n\) of these seeds and sowing them. He then carries out a hypothesis test at the 1% significance level to investigate whether he is correct.
  1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
  2. In a trial with \(n = 20\), Ramesh finds that 13 seeds germinate. Carry out the test. [4]
  3. Suppose instead that Ramesh conducts the trial with \(n = 50\), and finds that 33 seeds germinate. Given that the critical value for the test in this case is 35, complete the test. [3]
  4. If \(n\) is small, there is no point in carrying out the test at the 1% significance level, as the null hypothesis cannot be rejected however many seeds germinate. Find the least value of \(n\) for which the null hypothesis can be rejected, quoting appropriate probabilities to justify your answer. [3]
Edexcel S1 Q1
8 marks Moderate -0.8
An athlete believes that her times for running 200 metres in races are normally distributed with a mean of 22.8 seconds.
  1. Given that her time is over 23.3 seconds in 20\% of her races, calculate the variance of her times. [5]
  2. The record over this distance for women at her club is 21.82 seconds. According to her model, what is the chance that she will beat this record in her next race? [3]
Edexcel S1 Q2
10 marks Moderate -0.3
The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{5}{16}, \text{P}(B) = \frac{1}{2} \text{ and P}(A|B) = \frac{1}{4}$$ Find
  1. P\((A \cap B)\). [2]
  2. P\((B'|A)\). [3]
  3. P\((A' \cup B)\). [2]
  4. Determine, with a reason, whether or not the events \(A\) and \(B\) are independent. [3]
Edexcel S1 Q3
10 marks Moderate -0.3
A group of 60 children were each asked to choose an integer value between 1 and 9 inclusive. Their choices are summarised in the table below.
Value chosen123456789
Number of children345101213742
  1. Calculate the mean and standard deviation of the values chosen. [6]
It is suggested that the value chosen could be modelled by a discrete uniform distribution.
  1. Write down the mean that this model would predict. [2]
Given also that the standard deviation according to this model would be 2.58,
  1. explain why this model is not suitable and suggest why this is the case. [2]
Edexcel S1 Q4
13 marks Moderate -0.3
A six-sided die is biased such that there is an equal chance of scoring each of the numbers from 1 to 5 but a score of 6 is three times more likely than each of the other numbers.
  1. Write down the probability distribution for the random variable, \(X\), the score on a single throw of the die. [4]
  2. Show that E\((X) = \frac{33}{8}\). [3]
  3. Find E\((4X - 1)\). [2]
  4. Find Var\((X)\). [4]
Edexcel S1 Q5
17 marks Moderate -0.3
The number of patients attending a hospital trauma clinic each day was recorded over several months, giving the data in the table below.
Number of patients10 - 1920 - 2930 - 3435 - 3940 - 4445 - 4950 - 69
Frequency218243027145
These data are represented by a histogram. Given that the bar representing the 20 - 29 group is 2 cm wide and 7.2 cm high,
  1. calculate the dimensions of the bars representing the groups
    1. 30 - 34
    2. 50 - 69
    [6]
  2. Use linear interpolation to estimate the median and quartiles of these data. [6]
The lowest and highest numbers of patients recorded were 14 and 67 respectively.
  1. Represent these data with a boxplot drawn on graph paper and describe the skewness of the distribution. [5]
Edexcel S1 Q6
17 marks Moderate -0.3
Penshop have stores selling stationary in each of 6 towns. The population, \(P\), in tens of thousands and the monthly turnover, \(T\), in thousands of pounds for each of the shops are as recorded below.
TownAbbertonBemberClasterDellerEdgetonFigland
\(P\) (0.000's)3.27.65.29.08.14.8
\(T\) (£ 000's)11.112.413.319.317.911.8
  1. Represent these data on a scatter diagram with \(T\) on the vertical axis. [4]
    1. Which town's shop might appear to be underachieving given the populations of the towns?
    2. Suggest two other factors that might affect each shop's turnover. [3]
You may assume that $$\Sigma P = 37.9, \quad \Sigma T = 85.8, \quad \Sigma P^2 = 264.69, \quad \Sigma T^2 = 1286, \quad \Sigma PT = 574.25.$$
  1. Find the equation of the regression line of \(T\) on \(P\). [7]
  2. Estimate the monthly turnover that might be expected if a shop were opened in Gratton, a town with a population of 68 000. [2]
  3. Why might the management of Penshop be reluctant to use the regression line to estimate the monthly turnover they could expect if a shop were opened in Haggin, a town with a population of 172 000? [1]
Edexcel S1 Q1
6 marks Moderate -0.8
  1. Draw two separate scatter diagrams, each with eight points, to illustrate the relationship between \(x\) and \(y\) in the cases where they have a product moment correlation coefficient equal to
    1. exactly \(+1\),
    2. about \(-0.4\). [4 marks]
  2. Explain briefly how the conclusion you would draw from a product moment correlation coefficient of \(+0.3\) would vary according to the number of pairs of data used in its calculation. [2 marks]
Edexcel S1 Q2
6 marks Moderate -0.8
A histogram was drawn to show the distribution of age in completed years of the participants on an outward-bound course. There were 32 people aged 30-34 years on the course. The height of the rectangle representing this group was 19.2 cm and it was 1 cm in width. Given that there were 28 people aged 35-39 years,
  1. find the height of the rectangle representing this group. [3 marks]
Given that the height of the rectangle representing people aged 40-59 years was 2.7 cm,
  1. find the number of people on the course in this age group. [3 marks]
Edexcel S1 Q3
9 marks Moderate -0.3
The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{7}{12}, \quad \text{P}(A \cap B) = \frac{1}{4} \quad \text{and} \quad \text{P}(A|B) = \frac{2}{3}.$$ Find
  1. P\((B)\), [3 marks]
  2. P\((A \cup B)\), [3 marks]
  3. P\((B|A')\). [3 marks]
Edexcel S1 Q4
12 marks Standard +0.3
The owner of a mobile burger-bar believes that hot weather reduces his sales. To investigate the effect on his business he collected data on his daily sales, \(£P\), and the maximum temperature, \(T\)°C, on each of 20 days. He then coded the data, using \(x = T - 20\) and \(y = P - 300\), and calculated the summary statistics given below. $$\Sigma x = 57, \quad \Sigma y = 2222, \quad \Sigma x^2 = 401, \quad \Sigma y^2 = 305576, \quad \Sigma xy = 3871.$$
  1. Find an equation of the regression line of \(P\) on \(T\). [9 marks]
The owner of the bar doesn't believe it is profitable for him to run the bar if he takes less than £460 in a day.
  1. According to your regression line at what maximum daily temperature, to the nearest degree Celsius, does it become unprofitable for him to run the bar? [3 marks]
Edexcel S1 Q5
13 marks Moderate -0.8
The discrete random variable \(X\) has the probability function shown below. $$P(X = x) = \begin{cases} kx, & x = 2, 3, 4, 5, 6, \\ 0, & \text{otherwise}. \end{cases}$$
  1. Find the value of \(k\). [2 marks]
  2. Show that E\((X) = \frac{9}{2}\). [3 marks]
Find
  1. P\([X > \text{E}(X)]\), [2 marks]
  2. E\((2X - 5)\), [2 marks]
  3. Var\((X)\). [4 marks]
Edexcel S1 Q6
14 marks Standard +0.3
A geologist is analysing the size of quartz crystals in a sample of granite. She estimates that the longest diameter of 75% of the crystals is greater than 2 mm, but only 10% of the crystals have a longest diameter of more than 6 mm. The geologist believes that the distribution of the longest diameters of the quartz crystals can be modelled by a normal distribution.
  1. Find the mean and variance of this normal distribution. [9 marks]
The geologist also estimated that only 2% of the longest diameters were smaller than 1 mm.
  1. Calculate the corresponding percentage that would be predicted by a normal distribution with the parameters you calculated in part \((a)\). [3 marks]
  2. Hence, comment on the suitability of the normal distribution as a model in this situation. [2 marks]
Edexcel S1 Q7
15 marks Moderate -0.8
Jane and Tahira play together in a basketball team. The list below shows the number of points that Jane scored in each of 30 games.
39192830182123153424
29174312242541192640
45232132372418152436
  1. Construct a stem and leaf diagram for these data. [3 marks]
  2. Find the median and quartiles for these data. [4 marks]
  3. Represent these data with a boxplot. [3 marks]
Tahira played in the same 30 games and her lowest and highest points total in a game were 19 and 41 respectively. The quartiles for Tahira were 27, 31 and 35 respectively.
  1. Using the same scale draw a boxplot for Tahira's points totals. [2 marks]
  2. Compare and contrast the number of points scored per game by Jane and Tahira. [3 marks]
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]