Questions S1 (1967 questions)

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Edexcel S1 Q1
  1. \(70 \%\) of the households in a town have a CD player and \(45 \%\) have both a CD player and a personal computer (PC). 18\% have neither a CD player nor a PC.
    1. Illustrate this information using a Venn diagram.
    2. Find the percentage of the households that do not have a PC.
    3. Find the probability that a household chosen at random has a CD player or a PC but not both.
    4. The random variable \(X\) has the normal distribution \(\mathrm { N } \left( 2,1 \cdot 7 ^ { 2 } \right)\).
    5. State the standard deviation of \(X\).
    6. Find \(\mathrm { P } ( X < 0 )\).
    7. Find \(\mathrm { P } ( 0 \cdot 6 < X < 3 \cdot 4 )\).
    8. The discrete random variable \(X\) has probability function
    $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } c x ^ { 2 } & x = - 3 , - 2 , - 1,1,2,3
    0 & \text { otherwise. } \end{array} \right.$$
  2. Show that \(c = \frac { 1 } { 28 }\).
  3. Calculate
    1. \(\mathrm { E } ( X )\),
    2. \(\mathrm { E } \left( X ^ { 2 } \right)\).
  4. Calculate
    1. \(\operatorname { Var } ( X )\),
    2. \(\operatorname { Var } ( 10 - 2 X )\).
      (3 marks)
Edexcel S1 Q4
4. The heights, \(h \mathrm {~m}\), of eight children were measured, giving the following values of \(h\) : $$1 \cdot 20,1 \cdot 12,1 \cdot 43,0 \cdot 98,1 \cdot 31,1 \cdot 26,1 \cdot 02,1 \cdot 41$$
  1. Find the mean height of the children.
  2. Calculate the variance of the heights. The children were also weighed. It was found that their masses, \(w \mathrm {~kg}\), were such that $$\sum w = 324 , \quad \sum w ^ { 2 } = 13532 , \quad \sum w h = 403 .$$
  3. Calculate the product-moment correlation coefficient between \(w\) and \(h\).
  4. Comment briefly on the value you have obtained.
Edexcel S1 Q5
5. The ages of the residents of a retirement community are assumed to be normally distributed. \(15 \%\) of the residents are under 60 years old and \(5 \%\) are over 90 years old.
  1. Using this information, find the mean and the standard deviation of the ages.
  2. If there are 200 residents, find how many are over 80 years old. \section*{STATISTICS 1 (A) TEST PAPER 5 Page 2}
Edexcel S1 Q6
  1. Of the cars that are taken to a certain garage for an M.O.T. test, \(87 \%\) pass. However, \(2 \%\) of these have faults for which they should have been failed. \(5 \%\) of the cars which fail are in fact roadworthy and should have passed.
    Using a tree diagram, or otherwise, calculate the probabilities that a car chosen at random
    1. should have passed the test, regardless of whether it actually did or not,
    2. failed the test, given that it should have passed.
    The garage is told to improve its procedures. When it is inspected again a year later, it is found that the pass rate is still \(87 \%\) overall and \(2 \%\) of the cars passed have faults as before, but now \(0.3 \%\) of the cars which should have passed are failed and \(x \%\) of the cars which are failed should have passed.
  2. Find the value of \(x\).
Edexcel S1 Q7
7. The back-to-back stem and leaf diagram shows the journey times, to the nearest minute, of the commuter services into a big city provided by the trains of two operating companies.
Company \(A\)Company \(B\)
(3)4312056689(6)
(4)9865313479(5)
(4)8862401358()
(6)975321526899()
(3)65363477()
(3)3227015( )
Key : 4| 3 | 6 means 34 minutes for Company \(A\) and 36 minutes for Company \(B\).
  1. Write down the numbers needed to complete the diagram.
  2. Find the median and the quartiles for each company.
  3. On graph paper, draw box plots for the two companies. Show your scale.
  4. Use your plots to compare the two sets of data briefly.
  5. Describe the skewness of each company's distribution of times.
Edexcel S1 Q1
  1. (a) Briefly explain what is meant by a sample space.
    (b) State two properties which a function \(f ( x )\) must have to be a probability function.
  2. A company makes two cars, model \(A\) and model \(B\). The distance that model \(A\) travels on 10 litres of petrol is normally distributed with mean 109 km and variance \(72.25 \mathrm {~km} ^ { 2 }\). The distance that model \(B\) travels on 10 litres of petrol is normally distributed with mean 108.5 km and variance \(169 \mathrm {~km} ^ { 2 }\).
    In a trial, one of each model is filled with 10 litres of petrol and sent on a journey of 110 km . Find which model has the greater probability of completing this journey, and state the value of this probability.
  3. \(A , B\) and \(C\) are three events such that \(\mathrm { P } ( A ) = x , \mathrm { P } ( B ) = y\) and \(\mathrm { P } ( C ) = x + y\).
It is known that \(\mathrm { P } ( A \cup B ) = 0.6\) and \(\mathrm { P } ( B \mid A ) = 0.2\).
(a) Show that \(4 x + 5 y = 3\). It is also known that \(B\) and \(C\) are mutually exclusive and that \(\mathrm { P } ( B \cup C ) = 0.9\)
(b) Obtain another equation in \(x\) and \(y\) and hence find the values of \(x\) and \(y\).
(c) Deduce whether or not \(A\) and \(B\) are independent events.
Edexcel S1 Q4
4. The discrete random variable \(X\) has the following probability distribution :
\(x\)012345
\(\mathrm { P } ( X = x )\)0.110.170.20.13\(p\)\(p ^ { 2 }\)
  1. Find the value of \(p\).
  2. Find
    1. \(\mathrm { P } ( 0 < X \leq 2 )\),
    2. \(\mathrm { P } ( X \geq 3 )\).
  3. Find the mean and the variance of \(X\).
  4. Construct a table to represent the cumulative distribution function \(\mathrm { F } ( x )\).
Edexcel S1 Q5
5. The following marks out of 50 were given by two judges to the contestants in a talent contest:
Contestant\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Judge 1 \(( x )\)4332402147112938
Judge 2 \(( y )\)3925402236132732
Given that \(\sum x = 261 , \sum x ^ { 2 } = 9529\) and \(\sum x y = 8373\),
  1. calculate the product-moment correlation coefficient between the two judges' marks. \section*{STATISTICS 1 (A)TEST PAPER 6 Page 2} 5 continued...
  2. Find an equation of the regression line of \(x\) on \(y\). Contestant \(I\) was awarded 45 marks by Judge 2 .
  3. Estimate the mark that this contestant would have received from Judge 1.
  4. Comment, with explanation, on the probable accuracy of your answer.
Edexcel S1 Q6
6. 1000 houses were sold in a small town in a one-year period. The selling prices were as given in the following table:
Selling PriceNumber of HousesSelling PriceNumber of Houses
Up to \(\pounds 50000\)60Up to \(\pounds 150000\)642
Up to \(\pounds 75000\)227Up to \(\pounds 200000\)805
Up to \(\pounds 100000\)305Up to \(\pounds 500000\)849
Up to \(\pounds 125000\)414Up to \(\pounds 750000\)1000
  1. Name (do not draw) a suitable type of graph for illustrating this data.
  2. Use interpolation to find estimates of the median and the quartiles.
  3. Estimate the 37th percentile. Given further that the lowest price was \(\pounds 42000\) and the range of the prices was \(\pounds 690000\),
  4. draw a box plot to represent the data. Show your scale clearly. In another town the median price was \(\pounds 149000\), and the interquartile range was \(\pounds 90000\).
  5. Briefly compare the prices in the two towns using this information.
Edexcel S1 Q7
7. The random variable \(X\), which can take any value from \(\{ 1,2 , \ldots , n \}\), is modelled by the discrete uniform distribution with mean 10 .
  1. Show that \(n = 19\) and find the variance of \(X\).
  2. Find \(\mathrm { P } ( 3 < X \leq 6 )\). The random variable \(Y\) is defined by \(Y = 3 ( X - 10 )\).
  3. State the mean and the variance of \(Y\). The model for the distribution of \(X\) is found to be unsatisfactory, and in a refined model the probability distribution of \(X\) is taken to be $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k ( x + 1 ) & x = 1,2 , \ldots , 19
    0 & \text { otherwise } \end{array} \right.$$
  4. Show that \(k = \frac { 1 } { 209 }\).
  5. Find \(\mathrm { P } ( 3 < X \leq 6 )\) using this model.
Edexcel S1 Q1
  1. Twelve observations are made of a random variable \(X\). This set of observations has mean 13 and variance \(10 \cdot 2\).
Another twelve observations of \(X\) are such that \(\sum x = 164\) and \(\sum x ^ { 2 } = 2372\).
Find the mean and the variance for all twenty-four observations.
Edexcel S1 Q2
2. Given that \(\mathrm { P } ( A ) = \frac { 3 } { 5 } , \mathrm { P } ( B ) = \frac { 5 } { 8 } , \mathrm { P } ( A \cap B ) = \frac { 7 } { 20 } , \mathrm { P } ( A \cup C ) = \frac { 7 } { 10 }\) and \(\mathrm { P } ( C \mid A ) = \frac { 1 } { 3 }\),
  1. determine whether \(A\) and \(B\) are independent events.
  2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
  3. Find \(\mathrm { P } \left( ( A \cap C ) ^ { \prime } \right)\).
  4. Find \(\mathrm { P } ( A \mid C )\).
Edexcel S1 Q3
3. The frequency distribution for the lengths of 108 fish in an aquarium is given by the following table. The lengths of the fish ranged from 5 cm to 90 cm .
Length \(( \mathrm { cm } )\)\(5 - 10\)\(10 - 20\)\(20 - 25\)\(25 - 30\)\(30 - 40\)\(40 - 60\)\(60 - 90\)
Frequency8162018201412
  1. Calculate estimates of the three quartiles of the distribution.
  2. On graph paper, draw a box and whisker plot of the data.
  3. Hence describe the skewness of the distribution.
  4. If the data were represented by a histogram, what would be the ratio of the heights of the shortest and highest bars?
Edexcel S1 Q4
4. A botanist believes that the lengths of the branches on trees of a certain species can be modelled by a normal distribution.
When he measures the lengths of 500 branches, he finds 55 which are less than 30 cm long and 200 which are more than 90 cm long.
  1. Find the mean and the standard deviation of the lengths.
  2. In a sample of 1000 branches, how many would he expect to find with lengths greater than 1 metre? \section*{STATISTICS 1 (A) TEST PAPER 7 Page 2}
Edexcel S1 Q5
  1. Two spinners are in the form of an equilateral triangle, whose three regions are labelled 1,2 and 3, and a square, whose four regions are labelled \(1,2,3\) and 4 . Both spinners are biased and the probability distributions for the scores \(X\) and \(Y\) obtained when they are spun are respectively:
\(x\)123
\(\mathrm { P } ( X = x )\)\(0 \cdot 2\)\(0 \cdot 4\)\(p\)
\(Y\)1234
\(\mathrm { P } ( Y = y )\)0.20.5\(q\)\(q\)
  1. Find the values of \(p\) and \(q\).
  2. Find the probability that, when the two spinners are spun together, the sum of the two scores is (i) 5, (ii) less than 4 .
  3. State an assumption that you have made in answering part (b) and explain why it is likely to be justifiable.
  4. Calculate \(\mathrm { E } ( X + Y )\).
Edexcel S1 Q6
6. In a survey for a computer magazine, the times \(t\) seconds taken by eight laser printers to print a page of text were compared with the prices \(\pounds p\) of the printers. The data were coded using the equations \(x = t - 10\) and \(y = p - 150\), and it was found that $$\sum x = 42 \cdot 4 , \quad \sum x ^ { 2 } = 314 \cdot 5 , \quad \sum y = 560 , \quad \sum y ^ { 2 } = 60600 , \quad \sum x y = 1592 .$$
  1. Find the mean time and the mean price for the eight printers.
  2. Find the variance of the times.
  3. Find the equation of the regression line of \(p\) on \(t\).
  4. Estimate the price of a printer which takes 11.3 seconds to print the page.
Edexcel S1 Q1
  1. Using the coding \(y = \frac { x - 90 } { 5 }\), and showing each step in your working clearly, calculate the mean and the standard deviation of the 20 observations of a variable \(X\) given by the following table:
\(x\)7580859095100105110
Frequency12364211
Edexcel S1 Q2
  1. A darts player throws two darts, attempting to score a bull's-eye with each. The probability that he will achieve this with his first dart is 0.25 . If he misses with his first dart, the probability that he will also miss with his second dart is 0.7 . The probability that he will miss with at least one dart is 0.9 .
    1. Show that the probability that he succeeds with his first dart but misses with his second is 0.15 .
    2. Find the conditional probability that he misses with both darts, given that he misses with at least one.
    3. The entrance to a car park is 1.9 m wide. It is found that this is too narrow for \(2 \%\) of the vehicles which need to use the car park. The widths of these vehicles are modelled by a normal distribution with mean 1.6 m .
    4. Find the standard deviation of the distribution.
    It is decided to widen the entrance so that \(99.5 \%\) of vehicles will be able to use it.
  2. Find the minimum width needed to achieve this.
Edexcel S1 Q4
4. A pack of 52 cards contains 4 cards bearing each of the integers from 1 to 13 . A card is selected at random. The random variable \(X\) represents the number on the card.
  1. Find \(\mathrm { P } ( X \leq 5 )\).
  2. Name the distribution of \(X\) and find the expectation and variance of \(X\). A hand of 12 cards consists of three 2 s , four 3 s , two 4 s , two 5 s and one 6 . The random variable \(Y\) represents the number on a card chosen at random from this hand.
  3. Draw up a table to show the probability distribution of \(Y\).
  4. Calculate \(\operatorname { Var } ( 3 Y - 2 )\). \section*{STATISTICS 1 (A) TEST PAPER 8 Page 2}
Edexcel S1 Q5
  1. The following data were collected by counting the number of cars that passed the gates of a college in 60 successive 5 minute intervals.
122019313235372926272017
1598111317172125272825
303237404545444742413638
353430302726292423212118
161619222628231715101213
  1. Make a stem and leaf diagram for this data, using the groups \(5 - 9,10 - 14 , \ldots , 45 - 49\). Show the total in each group and give a key to the diagram.
  2. Find the three quartiles for this data.
  3. On graph paper, draw a box plot for the data.
  4. Describe the skewness of the distribution.
Edexcel S1 Q6
6. A missile was fired vertically upwards and its height above ground level, \(h\) metres, was found at various times \(t\) seconds after it was released. The results are given in the following table:
\(t\)1234567
\(h\)68126174216240252266
It is thought that this data can be fitted to the formula \(h = p t - q t ^ { 2 }\).
  1. Show that this equation can be written as \(\frac { h } { t } = p - q t\).
  2. Plot a scatter diagram of \(\frac { h } { t }\) against \(t\). Given that \(\sum h = 1342 , \sum \frac { h } { t } = 371\) and \(\sum \frac { h ^ { 2 } } { t ^ { 2 } } = 20385\),
  3. find the equation of the regression line of \(\frac { h } { t }\) on \(t\) and hence write down the values of \(p\) and \(q\).
  4. Use your equation to find the value of \(h\) when \(t = 10\). Comment on the implication of your answer.
  5. Find the product-moment correlation coefficient between \(\frac { h } { t }\) and \(t\) and state the significance of its value.
    (4 marks)
Edexcel S1 Q1
  1. Briefly describe what is meant by
    1. a statistical model,
    2. a refinement of a model.
    3. The random variable \(X\) has the discrete uniform distribution and takes the values \(\{ 1 , \ldots , n \}\). The standard deviation of of \(X\) is \(2 \sqrt { 6 }\). Find
    4. the mean of \(X\),
    5. \(\mathrm { P } \left( 3 \leq X < \frac { 1 } { 2 } n \right)\).
    6. The rainfall at a weather station was recorded every day of the twentieth century. One year is selected at random from the records and the total rainfall, in cm , in January of that year is denoted by \(R\) Assuming that \(R\) can be modelled by a normal distribution with standard deviation \(12 \cdot 6\), and given that \(\mathrm { P } ( R > 100 ) = 0 \cdot 0764\),
    7. find the mean of \(R\),
    8. calculate \(\mathrm { P } ( 75 < R < 80 )\).
    9. The length of time, in minutes, that visitors queued for a tourist attraction is given by the following table, where, for example, ' 20 - ' means from 20 up to but not including 30 minutes.
    Queuing time (mins)\(0 -\)\(10 -\)\(15 -\)\(20 -\)\(30 -\)\(40 - 60\)
    Number of visitors1524\(x\)1310\(y\)
  2. State the upper class boundary of the first class. A histogram is drawn to represent this data. The total area under the histogram is \(36 \mathrm {~cm} ^ { 2 }\). The ' 10 - ' bar has width 1 cm and height 9.6 cm . The ' 15 - ' bar is ten times as high as the '40-60' bar.
  3. Find the values of \(x\) and \(y\).
  4. On graph paper, construct the histogram accurately.
Edexcel S1 Q5
5. The discrete random variable \(X\) takes only the values 4, 5, 6, 7, 8 and 9. The probabilities of these values are given in the table:
\(x\)456789
\(\mathrm { P } ( X = x )\)\(p\)0.1\(q\)\(q\)0.30.2
It is known that \(\mathrm { E } ( X ) = 6 \cdot 7\). Find
  1. the values of \(p\) and \(q\),
  2. the value of \(a\) for which \(\mathrm { E } ( 2 X + a ) = 0\),
  3. \(\operatorname { Var } ( X )\). \section*{STATISTICS 1 (A) TEST PAPER 9 Page 2}
Edexcel S1 Q6
  1. The marks out of 75 obtained by a group of ten students in their first and second Statistics modules were as follows:
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Module 1 \(( x )\)54334271602739465964
Module 2 \(( y )\)50224458421935465560
  1. Find \(\sum x\) and \(\sum y\). Given that \(\sum x ^ { 2 } = 26353\) and \(\sum x y = 22991\),
  2. obtain the equation of the regression line of \(y\) on \(x\).
  3. Estimate the Module 2 result of a student whose mark in Module 1 was Explain why one of these estimates is less reliable than the other. The equation of the regression line of \(x\) on \(y\) is \(x = 0.921 y + 9.81\).
  4. Deduce the product moment correlation coefficient between \(x\) and \(y\), and briefly interpret its value.
Edexcel S1 Q7
7. Among the families with two children in a large city, the probability that the elder child is a boy is \(\frac { 5 } { 12 }\) and the probability that the younger child is a boy is \(\frac { 9 } { 16 }\). The probability that the younger child is a girl, given that the elder child is a girl, is \(\frac { 1 } { 4 }\).
One of the families is chosen at random. Using a tree diagram, or otherwise,
  1. show that the probability that both children are boys is \(\frac { 1 } { 8 }\). Find the probability that
  2. one child is a boy and the other is a girl,
  3. one child is a boy given that the other is a girl. If three of the families are chosen at random,
  4. find the probability that exactly two of the families have two boys.
  5. State an assumption that you have made in answering part (d).