Questions — Edexcel S1 (574 questions)

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Edexcel S1 2002 January Q1
  1. (a) Explain briefly what you understand by
    1. a statistical experiment,
    2. an event.
      (b) State one advantage and one disadvantage of a statistical model.
    3. A meteorologist measured the number of hours of sunshine, to the nearest hour, each day for 100 days. The results are summarised in the table below.
    Hours of sunshineDays
    116
    \(2 - 4\)32
    \(5 - 6\)28
    712
    89
    \(9 - 11\)2
    121
    (a) On graph paper, draw a histogram to represent these data.
    (b) Calculate an estimate of the number of days that had between 6 and 9 hours of sunshine.
Edexcel S1 2002 January Q3
3. A discrete random variable \(X\) has the probability function shown in the table below.
\(x\)012
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 3 }\)\(a\)\(\frac { 2 } { 3 } - a\)
  1. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 6 }\), find \(a\).
  2. Find the exact value of Var ( \(X\) ).
  3. Find the exact value of \(\mathrm { P } ( X \leq 15 )\).
Edexcel S1 2002 January Q4
4. A contractor bids for two building projects. He estimates that the probability of winning the first project is 0.5 , the probability of winning the second is 0.3 and the probability of winning both projects is 0.2 .
  1. Find the probability that he does not win either project.
  2. Find the probability that he wins exactly one project.
  3. Given that he does not win the first project, find the probability that he wins the second.
  4. By calculation, determine whether or not winning the first contract and winning the second contract are independent events.
Edexcel S1 2002 January Q5
5. The duration of the pregnancy of a certain breed of cow is normally distributed with mean \(\mu\) days and standard deviation \(\sigma\) days. Only \(2.5 \%\) of all pregnancies are shorter than 235 days and \(15 \%\) are longer than 286 days.
  1. Show that \(\mu - 235 = 1.96 \sigma\).
  2. Obtain a second equation in \(\mu\) and \(\sigma\).
  3. Find the value of \(\mu\) and the value of \(\sigma\).
  4. Find the values between which the middle \(68.3 \%\) of pregnancies lie.
Edexcel S1 2002 January Q6
6. Hospital records show the number of babies born in a year. The number of babies delivered by 15 male doctors is summarised by the stem and leaf diagram below.
Babies(4 5 means 45)Totals
0(0)
19(1)
21677(4)
322348(5)
45(1)
51(1)
60(1)
7(0)
867(2)
  1. Find the median and inter-quartile range of these data.
  2. Given that there are no outliers, draw a box plot on graph paper to represent these data. Start your scale at the origin.
  3. Calculate the mean and standard deviation of these data. The records also contain the number of babies delivered by 10 female doctors.
    343020156
    322619114
    The quartiles are 11, 19.5 and 30 .
  4. Using the same scale as in part (b) and on the same graph paper draw a box plot for the data for the 10 female doctors.
  5. Compare and contrast the box plots for the data for male and female doctors.
Edexcel S1 2002 January Q7
7. A number of people were asked to guess the calorific content of 10 foods. The
mean \(s\) of the guesses for each food and the true calorific content \(t\) are given in the table below.
Food\(t\)\(s\)
Packet of biscuits170420
1 potato90160
1 apple80110
Crisp breads1070
Chocolate bar260360
1 slice white bread75135
1 slice brown bread60115
Portion of beef curry270350
Portion of rice pudding165390
Half a pint of milk160200
[You may assume that \(\Sigma t = 1340 , \Sigma s = 2310 , \Sigma t s = 396775 , \Sigma t ^ { 2 } = 246050 , \Sigma s ^ { 2 } = 694650\).]
  1. Draw a scatter diagram, indicating clearly which is the explanatory (independent) and which is the response (dependent) variable.
  2. Calculate, to 3 significant figures, the product moment correlation coefficient for the above data.
  3. State, with a reason, whether or not the value of the product moment correlation coefficient changes if all the guesses are 50 calories higher than the values in the table. The mean of the guesses for the portion of rice pudding and for the packet of biscuits are outside the linear relation of the other eight foods.
  4. Find the equation of the regression line of \(s\) on \(t\) excluding the values for rice pudding and biscuits.
    [0pt] [You may now assume that \(S _ { t s } = 72587 , S _ { t t } = 63671.875 , \bar { t } = 125.625 , \bar { s } = 187.5\).]
  5. Draw the regression line on your scatter diagram.
  6. State, with a reason, what the effect would be on the regression line of including the values for a portion of rice pudding and a packet of biscuits. \section*{END}
Edexcel S1 2003 January Q1
  1. The total amount of time a secretary spent on the telephone in a working day was recorded to the nearest minute. The data collected over 40 days are summarised in the table below.
Time (mins)\(90 - 139\)\(140 - 149\)\(150 - 159\)\(160 - 169\)\(170 - 179\)\(180 - 229\)
No. of days81010444
Draw a histogram to illustrate these data
Edexcel S1 2003 January Q2
2. A car dealer offers purchasers a three year warranty on a new car. He sells two models, the Zippy and the Nifty. For the first 50 cars sold of each model the number of claims under the warranty is shown in the table below.
ClaimNo claim
Zippy3515
Nifty4010
One of the purchasers is chosen at random. Let \(A\) be the event that no claim is made by the purchaser under the warranty and \(B\) the event that the car purchased is a Nifty.
  1. Find \(\mathrm { P } ( A \cap B )\).
  2. Find \(\mathrm { P } \left( A ^ { \prime } \right)\). Given that the purchaser chosen does not make a claim under the warranty,
  3. find the probability that the car purchased is a Zippy.
  4. Show that making a claim is not independent of the make of the car purchased. Comment on this result.
Edexcel S1 2003 January Q3
3. A drinks machine dispenses coffee into cups. A sign on the machine indicates that each cup contains 50 ml of coffee. The machine actually dispenses a mean amount of 55 ml per cup and \(10 \%\) of the cups contain less than the amount stated on the sign. Assuming that the amount of coffee dispensed into each cup is normally distributed find
  1. the standard deviation of the amount of coffee dispensed per cup in ml ,
  2. the percentage of cups that contain more than 61 ml . Following complaints, the owners of the machine make adjustments. Only \(2.5 \%\) of cups now contain less than 50 ml . The standard deviation of the amount dispensed is reduced to 3 ml . Assuming that the amount of coffee dispensed is still normally distributed,
  3. find the new mean amount of coffee per cup.
    (4)
Edexcel S1 2003 January Q4
4. A restaurant owner is concerned about the amount of time customers have to wait before being served. He collects data on the waiting times, to the nearest minute, of 20 customers. These data are listed below.
15,14,16,15,17,16,15,14,15,16,
17,16,15,14,16,17,15,25,18,16
  1. Find the median and inter-quartile range of the waiting times. An outlier is an observation that falls either \(1.5 \times\) (inter-quartile range) above the upper quartile or \(1.5 \times\) (inter-quartile range) below the lower quartile.
  2. Draw a boxplot to represent these data, clearly indicating any outliers.
  3. Find the mean of these data.
  4. Comment on the skewness of these data. Justify your answer.
Edexcel S1 2003 January Q5
5. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) , & x = 0,1,2
k ( x - 2 ) , & x = 3
0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
  1. Show that \(k = 0.25\).
  2. Find \(\mathrm { E } ( X )\) and show that \(\mathrm { E } \left( X ^ { 2 } \right) = 2.5\).
  3. Find \(\operatorname { Var } ( 3 X - 2 )\). Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
  4. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\).
  5. Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
  6. Find \(\mathrm { P } \left( 1.3 \leq X _ { 1 } + X _ { 2 } \leq 3.2 \right)\).
Edexcel S1 2003 January Q6
6. The chief executive of Rex cars wants to investigate the relationship between the number of new car sales and the amount of money spent on advertising. She collects data from company records on the number of new car sales, \(c\), and the cost of advertising each year, \(p\) (£000). The data are shown in the table below.
YearNumber of new car sale, \(c\)Cost of advertising (£000), \(p\)
19904240120
19914380126
19924420132
19934440134
19944430137
19954520144
19964590148
19974660150
19984700153
19994790158
  1. Using the coding \(x = ( p - 100 )\) and \(y = \frac { 1 } { 10 } ( c - 4000 )\), draw a scatter diagram to represent these data. Explain why \(x\) is the explanatory variable.
  2. Find the equation of the least squares regression line of \(y\) on \(x\). $$\text { [Use } \left. \Sigma x = 402 , \Sigma y = 517 , \Sigma x ^ { 2 } = 17538 \text { and } \Sigma x y = 22611 . \right]$$
  3. Deduce the equation of the least squares regression line of \(c\) on \(p\) in the form \(c = a + b p\).
  4. Interpret the value of \(a\).
  5. Predict the number of extra new cars sales for an increase of \(\pounds 2000\) in advertising budget. Comment on the validity of your answer.
    (2)
Edexcel S1 2005 January Q1
  1. A company assembles drills using components from two sources. Goodbuy supplies \(85 \%\) of the components and Amart supplies the rest. It is known that \(3 \%\) of the components supplied by Goodbuy are faulty and \(6 \%\) of those supplied by Amart are faulty.
    1. Represent this information on a tree diagram.
    An assembled drill is selected at random.
  2. Find the probability that it is not faulty.
Edexcel S1 2005 January Q2
2. The number of caravans on Seaview caravan site on each night in August last year is summarised in the following stem and leaf diagram.
Caravans110 means 10Totals
10(2)
218(4)
30347(8)
41588(9)
5267(5)
62(3)
  1. Find the three quartiles of these data. During the same month, the least number of caravans on Northcliffe caravan site was 31. The maximum number of caravans on this site on any night that month was 72 . The three quartiles for this site were 38,45 and 52 respectively.
  2. On graph paper and using the same scale, draw box plots to represent the data for both caravan sites. You may assume that there are no outliers.
  3. Compare and contrast these two box plots.
  4. Give an interpretation to the upper quartiles of these two distributions.
Edexcel S1 2005 January Q3
3. The following table shows the height \(x\), to the nearest cm , and the weight \(y\), to the nearest kg , of a random sample of 12 students.
\(x\)148164156172147184162155182165175152
\(y\)395956774477654980727052
  1. On graph paper, draw a scatter diagram to represent these data.
  2. Write down, with a reason, whether the correlation coefficient between \(x\) and \(y\) is positive or negative. The data in the table can be summarised as follows. $$\Sigma x = 1962 , \quad \Sigma y = 740 , \quad \Sigma y ^ { 2 } = 47746 , \quad \Sigma x y = 122783 , \quad S _ { x x } = 1745 .$$
  3. Find \(S _ { x y }\). The equation of the regression line of \(y\) on \(x\) is \(y = - 106.331 + b x\).
  4. Find, to 3 decimal places, the value of \(b\).
  5. Find, to 3 significant figures, the mean \(\bar { y }\) and the standard deviation \(s\) of the weights of this sample of students.
  6. Find the values of \(\bar { y } \pm 1.96 s\).
  7. Comment on whether or not you think that the weights of these students could be modelled by a normal distribution.
Edexcel S1 2005 January Q4
4. The random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = k x , \quad x = 1,2 , \ldots , 5$$
  1. Show that \(k = \frac { 1 } { 15 }\). Find
  2. \(\mathrm { P } ( X < 4 )\),
  3. \(\mathrm { E } ( X )\),
  4. \(\mathrm { E } ( 3 X - 4 )\).
Edexcel S1 2005 January Q5
5. Articles made on a lathe are subject to three kinds of defect, \(A , B\) or \(C\). A sample of 1000 articles was inspected and the following results were obtained. \begin{displayquote} 31 had a type \(A\) defect
37 had a type \(B\) defect
42 had a type \(C\) defect
11 had both type \(A\) and type \(B\) defects
13 had both type \(B\) and type \(C\) defects
10 had both type \(A\) and type \(C\) defects
6 had all three types of defect.
  1. Draw a Venn diagram to represent these data. \end{displayquote} Find the probability that a randomly selected article from this sample had
  2. no defects,
  3. no more than one of these defects. An article selected at random from this sample had only one defect.
  4. Find the probability that it was a type \(B\) defect. Two different articles were selected at random from this sample.
  5. Find the probability that both had type \(B\) defects.
Edexcel S1 2005 January Q6
6. A discrete random variable is such that each of its values is assumed to be equally likely.
  1. Write down the name of the distribution that could be used to model this random variable.
  2. Give an example of such a distribution.
  3. Comment on the assumption that each value is equally likely.
  4. Suggest how you might refine the model in part (a).
Edexcel S1 2005 January Q7
7. The random variable \(X\) is normally distributed with mean 79 and variance 144 . Find
  1. \(\mathrm { P } ( X < 70 )\),
  2. \(\mathrm { P } ( 64 < X < 96 )\). It is known that \(\mathrm { P } ( 79 - a \leq X \leq 79 + b ) = 0.6463\). This information is shown in the figure below.
    \includegraphics[max width=\textwidth, alt={}, center]{df898ff4-c3ef-400c-b4f7-f4df3757941d-6_581_983_818_590} Given that \(\mathrm { P } ( X \geq 79 + b ) = 2 \mathrm { P } ( X \leq 79 - a )\),
  3. show that the area of the shaded region is 0.1179 .
  4. Find the value of \(b\).
Edexcel S1 2006 January Q1
  1. Over a period of time, the number of people \(x\) leaving a hotel each morning was recorded. These data are summarised in the stem and leaf diagram below.
Number leaving32 means 32Totals
2799(3)
322356(5)
401489(5)
5233666(7)
60145(4)
723(2)
81(1)
For these data,
  1. write down the mode,
  2. find the values of the three quartiles. Given that \(\Sigma x = 1335\) and \(\Sigma x ^ { 2 } = 71801\), find
  3. the mean and the standard deviation of these data. One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } } \text {. }$$
  4. Evaluate this measure to show that these data are negatively skewed.
  5. Give two other reasons why these data are negatively skewed.
Edexcel S1 2006 January Q2
2. The random variable \(X\) has probability distribution
\(x\)12345
\(\mathrm { P } ( X = x )\)0.10\(p\)0.20\(q\)0.30
  1. Given that \(\mathrm { E } ( X ) = 3.5\), write down two equations involving \(p\) and \(q\). Find
  2. the value of \(p\) and the value of \(q\),
  3. \(\operatorname { Var } ( X )\),
  4. \(\operatorname { Var } ( 3 - 2 X )\).
Edexcel S1 2006 January Q3
3. A manufacturer stores drums of chemicals. During storage, evaporation takes place. A random sample of 10 drums was taken and the time in storage, \(x\) weeks, and the evaporation loss, \(y \mathrm { ml }\), are shown in the table below.
\(x\)3568101213151618
\(y\)36505361697982908896
  1. On graph paper, draw a scatter diagram to represent these data.
  2. Give a reason to support fitting a regression model of the form \(y = a + b x\) to these data.
  3. Find, to 2 decimal places, the value of \(a\) and the value of \(b\). $$\text { (You may use } \Sigma x ^ { 2 } = 1352 , \Sigma y ^ { 2 } = 53112 \text { and } \Sigma x y = 8354 \text {.) }$$
  4. Give an interpretation of the value of \(b\).
  5. Using your model, predict the amount of evaporation that would take place after
    1. 19 weeks,
    2. 35 weeks.
  6. Comment, with a reason, on the reliability of each of your predictions.
Edexcel S1 2006 January Q4
4. A bag contains 9 blue balls and 3 red balls. A ball is selected at random from the bag and its colour is recorded. The ball is not replaced. A second ball is selected at random and its colour is recorded.
  1. Draw a tree diagram to represent the information. Find the probability that
  2. the second ball selected is red,
  3. both balls selected are red, given that the second ball selected is red.
Edexcel S1 2006 January Q5
5. (a) Write down two reasons for using statistical models.
(b) Give an example of a random variable that could be modelled by
  1. a normal distribution,
  2. a discrete uniform distribution.
Edexcel S1 2006 January Q6
6. For the events \(A\) and \(B\), $$\mathrm { P } \left( A \cap B ^ { \prime } \right) = 0.32 , \mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.11 \text { and } \mathrm { P } ( A \cup B ) = 0.65$$
  1. Draw a Venn diagram to illustrate the complete sample space for the events \(A\) and \(B\).
  2. Write down the value of \(\mathrm { P } ( A )\) and the value of \(\mathrm { P } ( B )\).
  3. Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
  4. Determine whether or not \(A\) and \(B\) are independent.