Questions — Edexcel S1 (574 questions)

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Edexcel S1 2004 November Q6
6. Students in Mr Brawn's exercise class have to do press-ups and sit-ups. The number of press-ups \(x\) and the number of sit-ups \(y\) done by a random sample of 8 students are summarised below. $$\begin{array} { l l } \Sigma x = 272 , & \Sigma x ^ { 2 } = 10164 , \quad \Sigma x y = 11222 ,
\Sigma y = 320 , & \Sigma y ^ { 2 } = 13464 . \end{array}$$
  1. Evaluate \(S _ { x x } , S _ { y y }\) and \(S _ { x y }\).
  2. Calculate, to 3 decimal places, the product moment correlation coefficient between \(x\) and \(y\).
  3. Give an interpretation of your coefficient.
  4. Calculate the mean and the standard deviation of the number of press-ups done by these students. Mr Brawn assumes that the number of press-ups that can be done by any student can be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). Assuming that \(\mu\) and \(\sigma\) take the same values as those calculated in part (d),
  5. find the value of \(a\) such that \(\mathrm { P } ( \mu - a < X < \mu + a ) = 0.95\).
  6. Comment on Mr Brawn's assumption of normality.
Edexcel S1 2004 November Q7
7. A college organised a 'fun run'. The times, to the nearest minute, of a random sample of 100 students who took part are summarised in the table below.
TimeNumber of students
\(40 - 44\)10
\(45 - 47\)15
4823
\(49 - 51\)21
\(52 - 55\)16
\(56 - 60\)15
  1. Give a reason to support the use of a histogram to represent these data.
  2. Write down the upper class boundary and the lower class boundary of the class 40-44.
  3. On graph paper, draw a histogram to represent these data. END
Edexcel S1 Specimen Q1
  1. (a) Explain what you understand by a statistical model.
    (2)
    (b) Write down a random variable which could be modelled by
    1. a discrete uniform distribution,
    2. a normal distribution.
    3. A group of students believes that the time taken to travel to college, \(T\) minutes, can be assumed to be normally distributed. Within the college \(5 \%\) of students take at least 55 minutes to travel to college and \(0.1 \%\) take less than 10 minutes.
    Find the mean and standard deviation of \(T\).
Edexcel S1 Specimen Q3
3. The discrete random variable \(X\) has probability function
\(\mathrm { P } ( X = x ) = \begin{cases} k x , & x = 1,2,3,4,5 ,
0 , & \text { otherwise } . \end{cases}\)
  1. Show that \(k = \frac { 1 } { 15 }\). Find the value of
  2. \(\mathrm { E } ( 2 X + 3 )\),
  3. \(\operatorname { Var } ( 2 X - 4 )\).
    (6)
Edexcel S1 Specimen Q4
4. A drilling machine can run at various speeds, but in general the higher the speed the sooner the drill needs to be replaced. Over several months, 15 pairs of observations relating to speed, \(s\) revolutions per minute, and life of drill, \(h\) hours, are collected. For convenience the data are coded so that \(x = s - 20\) and \(y = h - 100\) and the following summations obtained.
\(\Sigma x = 143 ; \Sigma y = 391 ; \Sigma x ^ { 2 } = 2413 ; \Sigma y ^ { 2 } = 22441 ; \Sigma x y = 484\).
  1. Find the equation of the regression line of \(h\) on \(s\).
  2. Interpret the slope of your regression line. Estimate the life of a drill revolving at 30 revolutions per minute.
    (2)
Edexcel S1 Specimen Q5
5. (a) Explain briefly the advantages and disadvantages of using the quartiles to summarise a set of data.
(b) Describe the main features and uses of a box plot. The distances, in kilometres, travelled to school by the teachers in two schools, \(A\) and \(B\), in the same town were recorded. The data for School \(A\) are summarised in Diagram 1. \section*{Diagram 1}
\includegraphics[max width=\textwidth, alt={}]{516911a4-d55e-4008-bad5-7c97bea94f9f-4_540_1244_772_390}
For School \(B\), the least distance travelled was 3 km and the longest distance travelled was 55 km . The three quartiles were 17, 24 and 31 respectively. An outlier is an observation that falls either \(1.5 \times\) (interquartile range) above the upper quartile or \(1.5 \times\) (interquartile range) below the lower quartile.
(c) Draw a box plot for School B.
(d) Compare and contrast the two box plots.
(4)
Edexcel S1 Specimen Q6
6. For any married couple who are members of a tennis club, the probability that the husband has a degree is \(\frac { 3 } { 5 }\) and the probability that the wife has a degree is \(\frac { 1 } { 2 }\). The probability that the husband has a degree, given that the wife has a degree, is \(\frac { 11 } { 12 }\). A married couple is chosen at random.
  1. Show that the probability that both of them have degrees is \(\frac { 11 } { 24 }\).
  2. Draw a Venn diagram to represent these data. Find the probability that
  3. only one of them has a degree,
  4. neither of them has a degree. Two married couples are chosen at random.
  5. Find the probability that only one of the two husbands and only one of the two wives have degrees.
Edexcel S1 Q1
  1. A histogram is to be drawn to represent the following grouped continuous data:
Group\(0 - 10\)\(10 - 20\)\(20 - 25\)\(25 - 30\)\(30 - 50\)\(50 - 100\)
Frequency\(2 x\)\(3 x\)\(5 x\)\(6 x\)\(2 x\)\(x\)
The ' \(10 - 20\) ' bar has height 6 cm and width 4 cm . Calculate
  1. the height of the ' \(20 - 25\) ' bar,
  2. the total area under the histogram.
Edexcel S1 Q2
2. The events \(A\) and \(B\) are independent. Given that \(\mathrm { P } ( A ) = 0.4\) and \(\mathrm { P } ( A \cap B ) = 0.12\), find
  1. \(\mathrm { P } ( B )\),
  2. \(\mathrm { P } ( A \cup B )\),
  3. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\),
  4. \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
Edexcel S1 Q3
3. The random variable \(X\) has the discrete uniform distribution over the set of consecutive integers \(\{ - 7 , - 6 , \ldots , 10 \}\).
Calculate (a) the expectation and variance of \(X\),
(b) \(\mathrm { P } ( X > 7 )\),
(c) the value of \(n\) for which \(\mathrm { P } ( - n \leq X \leq n ) = \frac { 7 } { 18 }\).
Edexcel S1 Q4
4. The marks, \(x\) out of 100 , scored by 30 candidates in an examination were as follows:
5192021232531373941
42444751565760616265
677071737577818298100
Given that \(\sum x = 1600\) and \(\sum x ^ { 2 } = 102400\),
  1. find the median, the mean and the standard deviation of these marks. The marks were scaled to give modified scores, \(y\), using the formula \(y = \frac { 4 x } { 5 } + 20\).
  2. Find the median, the mean and the standard deviation of the modified scores. \section*{STATISTICS 1 (A) TEST PAPER 1 Page 2}
Edexcel S1 Q5
  1. The table shows the numbers of cars and vans in a company's fleet having registrations with the prefix letters shown.
Registration letter\(K\)\(L\)\(M\)\(N\)\(P\)\(R\)\(S\)\(T\)\(V\)
Number of cars \(( x )\)67911151412107
Number of vans \(( y )\)810141313151498
  1. Plot a scatter graph of this data, with the number of cars on the horizontal axis and the number of vans on the vertical axis.
  2. If there were \(4 J\)-registered cars, estimate the number of \(J\)-registered vans. Given that \(\sum x ^ { 2 } = 1001 , \sum y ^ { 2 } = 1264\) and \(\sum x y = 1106\),
  3. calculate the product-moment correlation coefficient between \(x\) and \(y\). Give a brief interpretation of your answer.
Edexcel S1 Q6
6. The distributions of two independent discrete random variables \(X\) and \(Y\) are given in the tables:
\(x\)012
\(\mathrm { P } ( X = x )\)\(\frac { 3 } { 5 }\)\(\frac { 3 } { 10 }\)\(\frac { 1 } { 10 }\)
\(y\)01
\(\mathrm { P } ( Y = y )\)\(\frac { 5 } { 8 }\)\(\frac { 3 } { 8 }\)
The random variable \(Z\) is defined to be the sum of one observation from \(X\) and one from \(Y\).
  1. Tabulate the probability distribution for \(Z\).
  2. Calculate \(\mathrm { E } ( Z )\).
  3. Calculate (i) \(\mathrm { E } \left( Z ^ { 2 } \right)\), (ii) \(\operatorname { Var } ( Z )\).
  4. Calculate Var (3Z-4).
Edexcel S1 Q7
7. The times taken by a large number of people to read a certain book can be modelled by a normal distribution with mean \(5 \cdot 2\) hours. It is found that \(62 \cdot 5 \%\) of the people took more than \(4 \cdot 5\) hours to read the book.
  1. Show that the standard deviation of the times is approximately \(2 \cdot 2\) hours.
  2. Calculate the percentage of the people who took between 4 and 7 hours to read the book.
  3. Calculate the probability that two of the people chosen at random both took less than 5 hours to read the book, stating any assumption that you make.
  4. If a number of extra people were taken into account, all of whom took exactly \(5 \cdot 2\) hours to read the book, state with reasons what would happen to (i) the mean, (ii) the variance and explain briefly why the distribution would no longer be normal.
Edexcel S1 Q1
  1. (a) Explain briefly what is meant by a discrete random variable.
A family has 3 cats and 4 dogs. Two of the family's animals are to be chosen at random. The random variable \(X\) represents the number of dogs chosen.
(b) Copy and complete the table to show the probability distribution of \(X\) :
\(x\)012
\(\mathrm { P } ( X = x )\)
(c) Calculate
  1. \(\mathrm { E } ( X )\),
  2. \(\operatorname { Var } ( X )\),
  3. \(\operatorname { Var } ( 2 X )\).
Edexcel S1 Q2
2. The discrete random variable \(X\) can take any value in the set \(\{ 1,2,3,4,5,6,7,8 \}\). Arthur, Beatrice and Chris each carry out trials to investigate the distribution of \(X\). Arthur finds that \(\mathrm { P } ( X = 1 ) = 0.125\) and that \(\mathrm { E } ( X ) = 4.5\).
Beatrice finds that \(\mathrm { P } ( X = 2 ) = \mathrm { P } ( X = 3 ) = \mathrm { P } ( X = 4 ) = p\).
Chris finds that the values of \(X\) greater than 4 are all equally likely, with each having probability \(q\).
  1. Calculate the values of \(p\) and \(q\).
  2. Give the name for the distribution of \(X\).
  3. Calculate the standard deviation of \(X\).
Edexcel S1 Q3
3. The marks obtained by ten students in a Geography test and a History test were as follows:
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Geography \(( x )\)34574921845310776185
History \(( y )\)404955407139476573
  1. Given that \(\sum y = 547\), calculate the mark obtained by student \(E\) in History. Given further that \(\sum x ^ { 2 } = 34087 , \sum y ^ { 2 } = 31575\) and \(\sum x y = 31342\), calculate
  2. the product moment correlation coefficient between \(x\) and \(y\),
  3. an equation of the regression line of \(y\) on \(x\),
  4. an estimate of the History mark of student \(K\), who scored 70 in Geography.
  5. State, with a reason, whether you would expect your answer to part (d) to be reliable. \section*{STATISTICS 1 (A) TEST PAPER 2 Page 2}
Edexcel S1 Q4
  1. The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\).
    1. If \(2 \mu = 3 \sigma\), find \(\mathrm { P } ( X < 2 \mu )\).
    2. If, instead, \(\mathrm { P } ( X < 3 \mu ) = 0 \cdot 86\),
      1. find \(\mu\) in terms of \(\sigma\),
      2. calculate \(\mathrm { P } ( X > 0 )\).
    3. The stem-and-leaf diagram shows the values taken by two variables \(A\) and \(B\).
    \(A\)\(B\)
    \(8,7,4,1,0\)1\(1,1,2,5,6,8,9\)
    \(9,8,7,6,6,5,2\)2\(0,3,4,6,7,7,9\)
    \(9,7,6,4,2,1,0\)3\(1,4,5,5,8\)
    \(8,6,3,2,2\)4\(0,2,6,6,9,9\)
    \(6,4,0\)5\(2,3,5,7\)
    \(5,3,1\)60,1
    Key : 3| 1 | 2 means $$A = 13 , B = 12$$
  2. For each set of data, calculate estimates of the median and the quartiles.
  3. Calculate the 42nd percentile for \(A\).
  4. On graph paper, indicating your scale clearly, construct box and whisker plots for both sets of data.
  5. Describe the skewness of the distribution of \(A\) and of \(B\).
Edexcel S1 Q6
6. The values of the two variables \(A\) and \(B\) given in the table in Question 5 are written on cards and placed in two separate packs, which are labelled \(A\) and \(B\). One card is selected from Pack A. Let \(A _ { i }\) represent the event that the first digit on this card is \(i\).
  1. Write down the value of \(\mathrm { P } \left( A _ { 2 } \right)\). The card taken from Pack \(A\) is now transferred into Pack \(B\), and one card is picked at random from Pack \(B\). Let \(B _ { i }\) represent the event that the first digit on this card is \(i\).
  2. Show that \(\mathrm { P } \left( A _ { 1 } \cap B _ { 1 } \right) = \frac { 1 } { 24 }\).
  3. Show that \(\mathrm { P } \left( A _ { 6 } \mid B _ { 5 } \right) = \frac { 4 } { 41 }\).
  4. Find the value of \(\mathrm { P } \left( A _ { 1 } \cup B _ { 3 } \right)\).
Edexcel S1 Q1
  1. (a) Explain briefly what is meant by a random variable.
    (b) Write down a quantity which could be modelled as
    1. a discrete random variable,
    2. a continuous random variable.
    3. The discrete random variable \(X\) has the probability function given by the following table:
    \(x\)0123456
    \(\mathrm { P } ( X = x )\)0.090.120.220.16\(p\)\(2 p\)0.2
    (a) Show that \(p = 0.07\)
    (b) Find the value of \(\mathrm { E } ( X + 2 )\).
    (c) Find the value of \(\operatorname { Var } ( 3 X - 1 )\).
Edexcel S1 Q3
3. Twenty pairs of observations are made of two variables \(x\) and \(y\), which are believed to be related. It is found that $$\sum x = 200 , \quad \sum y = 174 , \quad \sum x ^ { 2 } = 6201 , \quad \sum y ^ { 2 } = 5102 , \quad \sum x y = 5200 .$$ Find
  1. the product-moment correlation coefficient between \(x\) and \(y\),
  2. the equation of the regression line of \(y\) on \(x\). Given that \(p = x + 30\) and \(q = y + 50\),
  3. find the equation of the regression line of \(q\) on \(p\), in the form \(q = m p + c\).
  4. Estimate the value of \(q\) when \(p = 46\), stating any assumptions you make.
Edexcel S1 Q4
4. The heights of the students at a university are assumed to follow a normal distribution. \(1 \%\) of the students are over 200 cm tall and 76\% are between 165 cm and 200 cm tall. Find
  1. the mean and the variance of the distribution,
  2. the percentage of the students who are under 158 cm tall.
  3. Comment briefly on the suitability of a normal distribution to model such a population. \section*{STATISTICS 1 (A) TEST PAPER 3 Page 2}
Edexcel S1 Q5
  1. In a survey of natural habitats, the numbers of trees in sixty equal areas of land were recorded, as follows:
171292340321153422318
154510521413294369301547
356241319269312718620
22183051493550258102631
332940373844243442381123
  1. Construct a stem-and-leaf diagram to illustrate this data, using the groupings 5-9, 10-14, 15-19, 20-24, etc.
  2. Find the three quartiles for the distribution.
  3. On graph paper construct a box plot for the data, showing your scale and clearly indicating any outliers.
Edexcel S1 Q6
6. Sixteen cards have been lost from a pack, which therefore contains only 36 cards. Two cards are drawn at random from the pack. The probability that both cards are red is \(\frac { 1 } { 3 }\).
  1. Show that \(r\), the number of red cards in the pack, satisfies the equation $$r ( r - 1 ) = 420$$
  2. Hence or otherwise find the value of \(r\).
  3. Find the probability that, when three cards are drawn at random from the pack,
    1. at least two are red,
    2. the first one is red given that at least two are red.
Edexcel S1 Q1
  1. Thirty cards, marked with the even numbers from 2 to 60 inclusive, are shuffled and one card is withdrawn at random and then replaced. The random variable \(X\) takes the value of the number on the card each time the experiment is repeated.
    1. What must be assumed about the cards if the distribution of \(X\) is modelled by a discrete uniform distribution?
    2. Making this modelling assumption, find the expectation and the variance of \(X\).
    3. (a) Explain briefly why, for data grouped in unequal classes, the class with the highest frequency may not be the modal class.
    In a histogram drawn to represent the annual incomes (in thousands of pounds) of 1000 families, the modal class was \(15 - 20\) (i.e. \(\mathrm { f } x\), where \(15000 \leq x < 20000\) ), with frequency 300 . The highest frequency in a class was 400 , for the class \(30 - 40\), and the bar representing this class was 8 cm high. The total area under the histogram was \(50 \mathrm {~cm} ^ { 2 }\).
  2. Find the height and the width of the bar representing the modal class.