Questions — Edexcel S1 (606 questions)

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Edexcel S1 2018 Specimen Q2
3 marks Moderate -0.8
  1. The time taken to complete a puzzle, in minutes, is recorded for each person in a club. The times are summarised in a grouped frequency distribution and represented by a histogram.
One of the class intervals has a frequency of 20 and is shown by a bar of width 1.5 cm and height 12 cm on the histogram. The total area under the histogram is \(94.5 \mathrm {~cm} ^ { 2 }\) Find the number of people in the club.
Edexcel S1 2018 Specimen Q3
14 marks Easy -1.2
3. The discrete random variable \(X\) has probability distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } \quad x = 1,2,3,4,5$$
  1. Write down the name given to this distribution. Find
  2. \(\mathrm { P } ( X = 4 )\)
  3. \(\mathrm { F } ( 3 )\)
  4. \(\mathrm { P } ( 3 X - 3 > X + 4 )\)
  5. Write down \(\mathrm { E } ( X )\)
  6. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
  7. Hence find \(\operatorname { Var } ( X )\) Given that \(\mathrm { E } ( a X - 3 ) = 11.4\)
  8. find \(\operatorname { Var } ( a X - 3 )\) \includegraphics[max width=\textwidth, alt={}, center]{b7500cc1-caa6-4767-bb2e-e3d70474e805-09_2261_54_312_34} \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
    VJYV SIHI NI JIIIM ION OC
    VJYV SIHI NI JLIYM ION OC
Edexcel S1 2018 Specimen Q4
12 marks Moderate -0.8
  1. A researcher recorded the time, \(t\) minutes, spent using a mobile phone during a particular afternoon, for each child in a club.
The researcher coded the data using \(v = \frac { t - 5 } { 10 }\) and the results are summarised in the table below.
Coded Time (v)Frequency ( \(\boldsymbol { f }\) )Coded Time Midpoint (m)
\(0 \leqslant v < 5\)202.5
\(5 \leqslant v < 10\)24\(a\)
\(10 \leqslant v < 15\)1612.5
\(15 \leqslant v < 20\)1417.5
\(20 \leqslant v < 30\)6\(b\)
$$\text { (You may use } \sum f m = 825 \text { and } \sum f m ^ { 2 } = 12012.5 \text { ) }$$
  1. Write down the value of \(a\) and the value of \(b\).
  2. Calculate an estimate of the mean of \(v\).
  3. Calculate an estimate of the standard deviation of \(v\).
  4. Use linear interpolation to estimate the median of \(v\).
  5. Hence describe the skewness of the distribution. Give a reason for your answer.
  6. Calculate estimates of the mean and the standard deviation of the time spent using a mobile phone during the afternoon by the children in this club. \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
    VJYV SIHI NI JIIIM ION OC
    VJYV SIHI NI JLIYM ION OC
Edexcel S1 2018 Specimen Q5
8 marks Moderate -0.3
  1. A biased tetrahedral die has faces numbered \(0,1,2\) and 3 . The die is rolled and the number face down on the die, \(X\), is recorded. The probability distribution of \(X\) is
\(x\)0123
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 2 }\)
If \(X = 3\) then the final score is 3
If \(X \neq 3\) then the die is rolled again and the final score is the sum of the two numbers.
The random variable \(T\) is the final score.
  1. Find \(\mathrm { P } ( T = 2 )\)
  2. Find \(\mathrm { P } ( T = 3 )\)
  3. Given that the die is rolled twice, find the probability that the final score is 3 \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
    VAYV SIHIL NI JIIIMM ION OC
    VJYV SIHI NI JLIYM ION OC
Edexcel S1 2018 Specimen Q6
11 marks Moderate -0.3
6. Three events \(A , B\) and \(C\) are such that $$\mathrm { P } ( A ) = \frac { 2 } { 5 } \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 8 }$$ Given that \(A\) and \(C\) are mutually exclusive find
  1. \(\mathrm { P } ( A \cup C )\) Given that \(A\) and \(B\) are independent
  2. show that \(\mathrm { P } ( B ) = \frac { 3 } { 8 }\)
  3. Find \(\mathrm { P } ( A \mid B )\) Given that \(\mathrm { P } \left( C ^ { \prime } \cap B ^ { \prime } \right) = 0.3\)
  4. draw a Venn diagram to represent the events \(A , B\) and \(C\) \includegraphics[max width=\textwidth, alt={}, center]{b7500cc1-caa6-4767-bb2e-e3d70474e805-21_2260_53_312_33} \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
    VJYV SIHI NI JIIIM ION OC
    VJYV SIHI NI JLIYM ION OC
Edexcel S1 2018 Specimen Q7
15 marks Moderate -0.3
  1. A machine fills bottles with water. The volume of water delivered by the machine to a bottle is \(X \mathrm { ml }\) where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)
One of these bottles of water is selected at random.
Given that \(\mu = 503\) and \(\sigma = 1.6\)
  1. find
    1. \(\mathrm { P } ( X > 505 )\)
    2. \(\mathrm { P } ( 501 < X < 505 )\)
  2. Find \(w\) such that \(\mathrm { P } ( 1006 - w < X < w ) = 0.9426\) Following adjustments to the machine, the volume of water delivered by the machine to a bottle is such that \(\mu = 503\) and \(\sigma = q\) Given that \(\mathrm { P } ( X < r ) = 0.01\) and \(\mathrm { P } ( X > r + 6 ) = 0.05\)
  3. find the value of \(r\) and the value of \(q\) \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
    VJYV SIHI NI JIIIM ION OC
    VEYV SIHI NI ELIYM ION OC
    VIAV SIHI NI BIIYM ION OOV34V SIHI NI IIIYM ION OOV38V SIHI NI JLIYM ION OC
Edexcel S1 Specimen Q1
4 marks Easy -1.2
  1. Gary compared the total attendance, \(x\), at home matches and the total number of goals, \(y\), scored at home during a season for each of 12 football teams playing in a league. He correctly calculated:
$$S _ { x x } = 1022500 \quad S _ { y y } = 130.9 \quad S _ { x y } = 8825$$
  1. Calculate the product moment correlation coefficient for these data.
  2. Interpret the value of the correlation coefficient. Helen was given the same data to analyse. In view of the large numbers involved she decided to divide the attendance figures by 100 . She then calculated the product moment correlation coefficient between \(\frac { x } { 100 }\) and \(y\).
  3. Write down the value Helen should have obtained.
Edexcel S1 Specimen Q2
10 marks Easy -1.2
2. An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability \(\frac { 2 } { 3 }\) of landing heads is spun.
When a blue ball is selected a fair coin is spun.
  1. Complete the tree diagram below to show the possible outcomes and associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-04_785_385_744_568} \includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-04_1054_483_760_954} Shivani selects a ball and spins the appropriate coin.
  2. Find the probability that she obtains a head. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,
  3. find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
  4. Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.
Edexcel S1 Specimen Q3
11 marks Moderate -0.8
  1. The discrete random variable \(X\) has probability distribution given by
\(x\)- 10123
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 5 }\)\(a\)\(\frac { 1 } { 10 }\)\(a\)\(\frac { 1 } { 5 }\)
where \(a\) is a constant.
  1. Find the value of \(a\).
  2. Write down \(\mathrm { E } ( X )\).
  3. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 6 - 2 X\)
  4. Find \(\operatorname { Var } ( Y )\).
  5. Calculate \(\mathrm { P } ( X \geqslant Y )\).
Edexcel S1 Specimen Q4
10 marks Moderate -0.8
  1. The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines \(A , B\) and \(C\).
\begin{figure}[h]
[diagram]
\captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} One of these students is selected at random.
  1. Show that the probability that the student reads more than one magazine is \(\frac { 1 } { 6 }\).
  2. Find the probability that the student reads \(A\) or \(B\) (or both).
  3. Write down the probability that the student reads both \(A\) and \(C\). Given that the student reads at least one of the magazines,
  4. find the probability that the student reads \(C\).
  5. Determine whether or not reading magazine \(B\) and reading magazine \(C\) are statistically independent.
Edexcel S1 Specimen Q5
14 marks Moderate -0.3
  1. A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week.
Hours\(1 - 10\)\(11 - 20\)\(21 - 25\)\(26 - 30\)\(31 - 40\)\(41 - 59\)
Frequency615111383
Mid-point5.515.52850
  1. Find the mid-points of the 21-25 hour and 31-40 hour groups. A histogram was drawn to represent these data. The 11-20 group was represented by a bar of width 4 cm and height 6 cm .
  2. Find the width and height of the 26-30 group.
  3. Estimate the mean and standard deviation of the time spent watching television by these students.
  4. Use linear interpolation to estimate the median length of time spent watching television by these students. The teacher estimated the lower quartile and the upper quartile of the time spent watching television to be 15.8 and 29.3 respectively.
  5. State, giving a reason, the skewness of these data.
Edexcel S1 Specimen Q6
14 marks Moderate -0.8
  1. A travel agent sells flights to different destinations from Beerow airport. The distance \(d\), measured in 100 km , of the destination from the airport and the fare \(\pounds f\) are recorded for a random sample of 6 destinations.
Destination\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)
\(d\)2.24.06.02.58.05.0
\(f\)182025233228
$$\text { [You may use } \sum d ^ { 2 } = 152.09 \quad \sum f ^ { 2 } = 3686 \quad \sum f d = 723.1 \text { ] }$$
  1. Using the axes below, complete a scatter diagram to illustrate this information.
  2. Explain why a linear regression model may be appropriate to describe the relationship between \(f\) and \(d\).
  3. Calculate \(S _ { d d }\) and \(S _ { f d }\)
  4. Calculate the equation of the regression line of \(f\) on \(d\) giving your answer in the form \(f = a + b d\).
  5. Give an interpretation of the value of \(b\). Jane is planning her holiday and wishes to fly from Beerow airport to a destination \(t \mathrm {~km}\) away. A rival travel agent charges 5 p per km.
  6. Find the range of values of \(t\) for which the first travel agent is cheaper than the rival. \includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-20_967_1630_1722_164}
Edexcel S1 Specimen Q7
12 marks Moderate -0.3
  1. The distances travelled to work, \(D \mathrm {~km}\), by the employees at a large company are normally distributed with \(D \sim \mathrm {~N} \left( 30,8 ^ { 2 } \right)\).
    1. Find the probability that a randomly selected employee has a journey to work of more than 20 km .
    2. Find the upper quartile, \(Q _ { 3 }\), of \(D\).
    3. Write down the lower quartile, \(Q _ { 1 }\), of \(D\).
    An outlier is defined as any value of \(D\) such that \(D < h\) or \(D > k\) where $$h = Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \quad \text { and } \quad k = Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
  2. Find the value of \(h\) and the value of \(k\). An employee is selected at random.
  3. Find the probability that the distance travelled to work by this employee is an outlier.
    END
Edexcel S1 2001 January Q1
7 marks Easy -1.3
  1. The students in a class were each asked to write down how many CDs they owned. The student with the least number of CDs had 14 and all but one of the others owned 60 or fewer. The remaining student owned 65 . The quartiles for the class were 30,34 and 42 respectively.
Outliers are defined to be any values outside the limits of \(1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) below the lower quartile or above the upper quartile. On graph paper draw a box plot to represent these data, indicating clearly any outliers.
marks)
Edexcel S1 2001 January Q2
8 marks Easy -1.2
2. The random variable \(X\) is normally distributed with mean 177.0 and standard deviation 6.4.
  1. Find \(\mathrm { P } ( 166 < X < 185 )\). It is suggested that \(X\) might be a suitable random variable to model the height, in cm , of adult males.
  2. Give two reasons why this is a sensible suggestion.
  3. Explain briefly why mathematical models can help to improve our understanding of real-world problems.
Edexcel S1 2001 January Q3
12 marks Easy -1.2
3. A fair six-sided die is rolled. The random variable \(Y\) represents the score on the uppermost, face.
  1. Write down the probability function of \(Y\).
  2. State the name of the distribution of \(Y\). Find the value of
  3. \(\mathrm { E } ( 6 Y + 2 )\),
  4. \(\operatorname { Var } ( 4 Y - 2 )\).
Edexcel S1 2001 January Q4
13 marks Easy -1.3
4. The employees of a company are classified as management, administration or production. The following table shows the number employed in each category and whether or not they live close to the company or some distance away.
Live close
Live some
distance away
Management614
Administration2510
Production4525
An employee is chosen at random.
Find the probability that this employee
  1. is an administrator,
  2. lives close to the company, given that the employee is a manager. Of the managers, \(90 \%\) are married, as are \(60 \%\) of the administrators and \(80 \%\) of the production employees.
  3. Construct a tree diagram containing all the probabilities.
  4. Find the probability that an employee chosen at random is married. An employee is selected at random and found to be married.
  5. Find the probability that this employee is in production.
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)