Questions — Edexcel (10514 questions)

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Edexcel S1 2001 January Q5
17 marks Moderate -0.3
5. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 200 motorists were delayed by roadworks on a stretch of motorway.
Delay (mins)Number of motorists
\(4 - 6\)15
\(7 - 8\)28
949
1053
\(11 - 12\)30
\(13 - 15\)15
\(16 - 20\)10
  1. Using graph paper represent these data by a histogram.
  2. Give a reason to justify the use of a histogram to represent these data.
  3. Use interpolation to estimate the median of this distribution.
  4. Calculate an estimate of the mean and an estimate of the standard deviation of these data. One coefficient of skewness is given by $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
  5. Evaluate this coefficient for the above data.
  6. Explain why the normal distribution may not be suitable to model the number of minutes that motorists are delayed by these roadworks.
Edexcel S1 2001 January Q6
18 marks Moderate -0.8
6. A local authority is investigating the cost of reconditioning its incinerators. Data from 10 randomly chosen incinerators were collected. The variables monitored were the operating time \(x\) (in thousands of hours) since last reconditioning and the reconditioning cost \(y\) (in \(\pounds 1000\) ). None of the incinerators had been used for more than 3000 hours since last reconditioning. The data are summarised below, $$\Sigma x = 25.0 , \Sigma x ^ { 2 } = 65.68 , \Sigma y = 50.0 , \Sigma y ^ { 2 } = 260.48 , \Sigma x y = 130.64 .$$
  1. Find \(\mathrm { S } _ { x x } , \mathrm {~S} _ { x y } , \mathrm {~S} _ { y y }\).
  2. Calculate the product moment correlation coefficient between \(x\) and \(y\).
  3. Explain why this value might support the fitting of a linear regression model of the form \(y = a + b x\).
  4. Find the values of \(a\) and \(b\).
  5. Give an interpretation of \(a\).
  6. Estimate
    1. the reconditioning cost for an operating time of 2400 hours,
    2. the financial effect of an increase of 1500 hours in operating time.
  7. Suggest why the authority might be cautious about making a prediction of the reconditioning cost of an incinerator which had been operating for 4500 hours since its last reconditioning.
Edexcel S1 2003 January Q1
4 marks Easy -1.3
  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
9 marks Easy -1.2
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
11 marks Standard +0.3
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
16 marks Easy -1.2
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
16 marks Standard +0.3
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
19 marks Moderate -0.3
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
6 marks Easy -1.3
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
14 marks Easy -1.8
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
15 marks Easy -1.3
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
8 marks Easy -1.2
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
13 marks Moderate -0.8
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 marks Easy -1.2
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
13 marks Standard +0.3
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
14 marks Easy -1.3
  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
12 marks Moderate -0.8
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
18 marks Easy -1.2
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
7 marks Easy -1.2
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
    1. the second ball selected is red,
    2. both balls selected are red, given that the second ball selected is red.
Edexcel S1 2006 January Q5
4 marks Easy -1.8
5.
  1. Write down two reasons for using statistical models.
  2. 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
11 marks Standard +0.3
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.
Edexcel S1 2006 January Q7
9 marks Moderate -0.8
7. The heights of a group of athletes are modelled by a normal distribution with mean 180 cm and a standard deviation 5.2 cm . The weights of this group of athletes are modelled by a normal distribution with mean 85 kg and standard deviation 7.1 kg . Find the probability that a randomly chosen athlete
  1. is taller than 188 cm ,
  2. weighs less than 97 kg .
    (2)
  3. Assuming that for these athletes height and weight are independent, find the probability that a randomly chosen athlete is taller than 188 cm and weighs more than 97 kg .
  4. Comment on the assumption that height and weight are independent.
Edexcel S1 2007 January Q1
15 marks Moderate -0.8
  1. As part of a statistics project, Gill collected data relating to the length of time, to the nearest minute, spent by shoppers in a supermarket and the amount of money they spent. Her data for a random sample of 10 shoppers are summarised in the table below, where \(t\) represents time and \(\pounds m\) the amount spent over \(\pounds 20\).
\(t\) (minutes)£m
15-3
2317
5-19
164
3012
6-9
3227
236
3520
276
  1. Write down the actual amount spent by the shopper who was in the supermarket for 15 minutes.
  2. Calculate \(S _ { t t } , S _ { m m }\) and \(S _ { t m }\). $$\text { (You may use } \Sigma t ^ { 2 } = 5478 \Sigma m ^ { 2 } = 2101 \Sigma t m = 2485 \text { ) }$$
  3. Calculate the value of the product moment correlation coefficient between \(t\) and \(m\).
  4. Write down the value of the product moment correlation coefficient between \(t\) and the actual amount spent. Give a reason to justify your value. On another day Gill collected similar data. For these data the product moment correlation coefficient was 0.178
  5. Give an interpretation to both of these coefficients.
  6. Suggest a practical reason why these two values are so different.
Edexcel S1 2007 January Q2
11 marks Moderate -0.8
In a factory, machines \(A , B\) and \(C\) are all producing metal rods of the same length. Machine \(A\) produces \(35 \%\) of the rods, machine \(B\) produces \(25 \%\) and the rest are produced by machine \(C\). Of their production of rods, machines \(A , B\) and \(C\) produce \(3 \% , 6 \%\) and \(5 \%\) defective rods respectively.
  1. Draw a tree diagram to represent this information.
  2. Find the probability that a randomly selected rod is
    1. produced by machine \(A\) and is defective,
    2. is defective.
  3. Given that a randomly selected rod is defective, find the probability that it was produced by machine \(C\).
Edexcel S1 2007 January Q3
13 marks Moderate -0.8
  1. The random variable \(X\) has probability function
$$\mathrm { P } ( X = x ) = \frac { ( 2 x - 1 ) } { 36 } \quad x = 1,2,3,4,5,6$$
  1. Construct a table giving the probability distribution of \(X\). Find
  2. \(\mathrm { P } ( 2 < X \leqslant 5 )\),
  3. the exact value of \(\mathrm { E } ( X )\).
  4. Show that \(\operatorname { Var } ( X ) = 1.97\) to 3 significant figures.
  5. Find \(\operatorname { Var } ( 2 - 3 X )\).