Questions — Edexcel S1 (606 questions)

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Edexcel S1 Q2
11 marks Moderate -0.8
2. The discrete random variable \(X\) has the probability function shown below. $$P ( X = x ) = \left\{ \begin{array} { l c } \frac { k } { x } , & x = 1,2,3,4 \\ 0 , & \text { otherwise } . \end{array} \right.$$
  1. Show that \(k = \frac { 12 } { 25 }\) Find
  2. \(\mathrm { F } ( 2 )\),
  3. \(\mathrm { E } ( X )\),
  4. \(\mathrm { E } \left( X ^ { 2 } + 2 \right)\).
Edexcel S1 Q3
12 marks Standard +0.3
3. A study was made of the heights of boys of different ages in Lancashire. The study concluded that the heights of 13 year-old boys are normally distributed with a mean of 156 cm and a variance of \(73 \mathrm {~cm} ^ { 2 }\). Find the probability that a 13 year-old boy chosen at random will be
  1. more than 165 cm tall,
  2. between 156 and 165 cm tall. The study also concluded that the heights of 14 year-old boys are normally distributed with a mean of 160 cm and a variance of \(79 \mathrm {~cm} ^ { 2 }\). One 13 year-old and one 14 year-old boy are chosen at random.
  3. Find the probability that both boys are more than 165 cm tall.
  4. State, with a reason, whether the probability that the combined height of the two boys is more than 330 cm is more or less than your answer to part (c).
    (2 marks)
Edexcel S1 Q4
13 marks Moderate -0.8
4. A company offering a bicycle courier service within London collected data on the delivery times for a sample of jobs completed by staff at each of its two offices. The times, \(t\) minutes, for 20 deliveries handled by the company's Hammersmith office were summarised by $$\Sigma t = 427 , \text { and } \Sigma t ^ { 2 } = 11077$$
  1. Find the mean and variance of the delivery times in this sample. The company's Holborn office handles more business, so the delivery times for a sample of 30 jobs handled by this office was taken. The mean and standard deviation of this sample were 18.5 minutes and 8.2 minutes respectively.
  2. Find the mean and variance of the delivery times of the combined sample of 50 deliveries.
Edexcel S1 Q5
14 marks Moderate -0.8
5. A College employs 75 teachers, of whom 47 are full-time and the rest are part-time. Of the 39 male teachers at the College, 26 are full-time.
  1. Represent this information on a Venn diagram.
  2. One teacher is selected at random to be interviewed by an inspector. Find the probability that the teacher chosen
    1. works full-time and is female,
    2. works part-time, given that he is male.
  3. Three teachers are selected at random to be observed by an inspector during one day. Find correct to 3 significant figures the probability that
    1. all three teachers chosen work full-time,
    2. at least one of the three teachers chosen is female.
Edexcel S1 Q6
14 marks Moderate -0.8
6. A physics student recorded the length, \(l \mathrm {~cm}\), of a spring when different masses, \(m\) grams, were suspended from it giving the following results.
\(m ( \mathrm {~g} )\)50100200300400500600700
\(l ( \mathrm {~cm} )\)7.810.716.522.128.033.935.235.6
  1. Represent these data on a scatter diagram with \(l\) on the vertical axis. The student decides to find the equation of a regression line of the form \(l = a + b m\) using only the data for \(m \leq 500 \mathrm {~g}\).
  2. Give a reason to support the fitting of such a regression line and explain why the student is excluding two of his values.
    (2 marks)
    You may use $$\Sigma m = 1550 , \quad \Sigma l = 119 , \quad \Sigma m ^ { 2 } = 552500 , \quad \Sigma l ^ { 2 } = 2869.2 , \quad \Sigma m l = 39540 .$$
  3. Find the values of \(a\) and \(b\).
  4. Explain the significance of the values of \(a\) and \(b\) in this situation.
Edexcel S1 Q1
10 marks Moderate -0.8
  1. There are 16 competitors in a table-tennis competition, 5 of which come from Racknor Comprehensive School. Prizes are awarded to the competitors finishing in each of first, second and third place.
Assuming that all the competitors have an equal chance of success, find the probability that the students from Racknor Comprehensive
  1. win no prizes,
  2. win the \(1 ^ { \text {st } }\) and \(3 ^ { \text {rd } }\) place prizes but not the \(2 ^ { \text {nd } }\) place prize,
  3. win exactly one of the prizes.
Edexcel S1 Q2
10 marks Moderate -0.3
2. A statistics student gave a questionnaire to a random sample of 50 pupils at his school. The sample included pupils aged from 11 to 18 years old. The student summarised the data on age in completed years, \(A\), and the number of hours spent doing homework in the previous week, \(H\), giving the following: $$\Sigma A = 703 , \quad \Sigma H = 217 , \quad \Sigma A ^ { 2 } = 10131 , \quad \Sigma H ^ { 2 } = 1338.5 , \quad \Sigma A H = 3253.5$$
  1. Calculate the product moment correlation coefficient for these data and explain what is shown by your result.
    (6 marks)
    The student also asked each pupil how many hours of paid work they had done in the previous week. He then calculated the product moment correlation coefficient for the data on hours doing homework and hours doing paid work, giving a value of \(r = 0.5213\) The student concluded that paid work did not interfere with homework as pupils doing more paid work also tended to do more homework.
  2. Explain why this conclusion may not be valid.
  3. Explain briefly how the student could more effectively investigate the effect of paid work on homework.
    (2 marks)
Edexcel S1 Q3
11 marks Moderate -0.3
3. A soccer fan collected data on the number of minutes of league football, \(m\), played by each team in the four main divisions before first scoring a goal at the start of a new season. Her results are shown in the table below.
\(m\) (minutes)Number of teams
\(0 \leq m < 40\)36
\(40 \leq m < 80\)28
\(80 \leq m < 120\)10
\(120 \leq m < 160\)4
\(160 \leq m < 200\)5
\(200 \leq m < 300\)4
\(300 \leq m < 400\)2
\(400 \leq m < 600\)3
  1. Calculate estimates of the mean and standard deviation of these data.
  2. Explain why the mean and standard deviation might not be the best summary statistics to use with these data.
  3. Suggest alternative summary statistics that would better represent these data.
Edexcel S1 Q4
13 marks Standard +0.3
4. Alan runs on a treadmill each day for as long as he can at 7 miles per hour. The length of time for which he runs is normally distributed with a mean of 21.6 minutes and a standard deviation of 1.8 minutes.
  1. Calculate the probability that on any one day Alan will run for less than 20 minutes.
  2. Estimate the number of times in a ninety-day period that Alan will run for more than 24 minutes.
  3. On a particular day Alan is still running after 22 minutes. Find the probability that he will stop running in the next 2 minutes.
Edexcel S1 Q5
14 marks Easy -1.2
5. In a survey unemployed people were asked how many months it had been, to the nearest month, since they were last employed on a full-time basis. The data collected is summarised in this stem and leaf diagram.
Number of months(2 | 1 means 21 months)Totals
011224446779(11)
102355689( )
21568( )
3079( )
45( )
527(2)
63(1)
70(1)
  1. Write down the values needed to complete the totals column on the stem and leaf diagram.
  2. State the mode of these data.
  3. Find the median and quartiles of these data. Given that any values outside of the limits \(\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) are to be regarded as outliers,
  4. determine if there are any outliers in these data,
  5. draw a box plot representing these data on graph paper,
  6. describe the skewness of these data and suggest a reason for it.
Edexcel S1 Q6
17 marks Easy -1.8
6. In a game two spinners are used. The score on the first spinner is given by the random variable \(A\), which has the following probability distribution:
\(a\)123
\(\mathrm { P } ( A = a )\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)
  1. State the name of this distribution.
  2. Write down \(\mathrm { E } ( A )\). The score on the second spinner is given by the random variable \(B\), which has the following probability distribution:
    \(b\)123
    \(\mathrm { P } ( B = b )\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 4 }\)
  3. Find \(\mathrm { E } ( B )\). On each player's turn in the game, both spinners are used and the scores on the two spinners are added together. The total score on the two spinners is given by the random variable \(C\).
  4. Show that \(\mathrm { P } ( C = 2 ) = \frac { 1 } { 6 }\).
  5. Find the probability distribution of \(C\).
  6. Show that \(\mathrm { E } ( C ) = \mathrm { E } ( A ) + \mathrm { E } ( B )\).
Edexcel S1 Q1
7 marks Moderate -0.8
  1. A shop recorded the number of pairs of gloves, \(n\), that it sold and the average daytime temperature, \(T ^ { \circ } \mathrm { C }\), for each month over a 12-month period.
The data was then summarised as follows: $$\Sigma T = 124 , \quad \Sigma n = 384 , \quad \Sigma T ^ { 2 } = 1802 , \quad \Sigma n ^ { 2 } = 18518 , \quad \Sigma T n = 2583 .$$
  1. Calculate the product moment correlation coefficient for these data.
  2. Comment on what your value shows and suggest a reason for this.
Edexcel S1 Q2
8 marks Standard +0.3
2. Events \(A\) and \(B\) are independent. Given also that $$\mathrm { P } ( A ) = \frac { 3 } { 4 } \quad \text { and } \quad \mathrm { P } \left( A \cap B ^ { \prime } \right) = \frac { 1 } { 4 }$$ Find
  1. \(\mathrm { P } ( A \cap B )\),
  2. \(\mathrm { P } ( B )\),
  3. \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)\).
Edexcel S1 Q3
10 marks Easy -1.2
3. The random variable \(X\) is such that $$\mathrm { E } ( X ) = a \text { and } \operatorname { Var } ( X ) = b$$ Find expressions in terms of \(a\) and \(b\) for
  1. \(\mathrm { E } ( 2 X + 3 )\),
  2. \(\quad \operatorname { Var } ( 2 X + 3 )\),
  3. \(\mathrm { E } \left( X ^ { 2 } \right)\).
  4. Show that $$\mathrm { E } \left[ ( X + 1 ) ^ { 2 } \right] = ( a + 1 ) ^ { 2 } + b$$
Edexcel S1 Q4
11 marks Standard +0.3
  1. An engineer tested a new material under extreme conditions in a wind tunnel. He recorded the number of microfractures, \(n\), that formed and the wind speed, \(v\) metres per second, for 8 different values of \(v\) with all other conditions remaining constant. He then coded the data using \(x = v - 700\) and \(y = n - 20\) and calculated the following summary statistics.
$$\Sigma x = 100 , \quad \Sigma y = 23 , \quad \Sigma x ^ { 2 } = 215000 , \quad \Sigma x y = 11600 .$$
  1. Find an equation of the regression line of \(y\) on \(x\).
  2. Hence, find an equation of the regression line of \(n\) on \(v\).
  3. Use your regression line to estimate the number of microfractures that would be formed if the material was tested in a wind speed of 900 metres per second with all other conditions remaining constant.
    (2 marks)
Edexcel S1 Q5
12 marks Moderate -0.3
5. An antiques shop recorded the value of items stolen to the nearest pound during each week for a year giving the data in the table below.
Value of goods stolen (£)Number of weeks
0-19931
200-3996
400-5993
600-7994
800-9995
1000-19992
2000-29991
Letting \(x\) represent the mid-point of each group and using the coding \(y = \frac { x - 699.5 } { 200 }\),
  1. find \(\sum\) fy.
  2. estimate to the nearest pound the mean and standard deviation of the value of the goods stolen each week using your value for \(\sum f y\) and \(\sum f y ^ { 2 } = 424\).
    (6 marks)
    The median for these data is \(\pounds 82\).
  3. Explain why the manager of the shop might be reluctant to use either the mean or the median in summarising these data.
    (3 marks)
Edexcel S1 Q6
12 marks Moderate -0.3
6. At the start of a gameshow there are 10 contestants of which 6 are female. In each round of the game, one contestant is eliminated. All of the contestants have the same chance of progressing to the next round each time.
  1. Show that the probability that the first two contestants to be eliminated are both male is \(\frac { 2 } { 15 }\).
  2. Find the probability that more females than males are eliminated in the first three rounds of the game.
  3. Given that the first contestant to be eliminated is male, find the probability that the next two contestants to be eliminated are both female.
    (3 marks)
Edexcel S1 Q7
15 marks Moderate -0.8
7. A cyber-cafe recorded how long each user stayed during one day giving the following results.
Length of stay
(minutes)
\(0 -\)\(30 -\)\(60 -\)\(90 -\)\(120 -\)\(240 -\)\(360 -\)
Number of users153132231720
  1. Use linear interpolation to estimate the median and quartiles of these data. The results of a previous study had led to the suggestion that the length of time each user stays can be modelled by a normal distribution with a mean of 72 minutes and a standard deviation of 48 minutes.
  2. Find the median and quartiles that this model would predict.
  3. Comment on the suitability of the suggested model in the light of the new results.
Edexcel S1 Q2
9 marks Moderate -0.8
  1. Plot a scatter diagram showing these data. The student wanted to investigate further whether or not her data provided evidence of an increase in temperature in June each year. Using \(Y\) for the number of years since 1993 and \(T\) for the mean temperature, she calculated the following summary statistics. $$\Sigma Y = 28 , \quad \Sigma T = 182.5 , \quad \Sigma Y ^ { 2 } = 140 , \quad \Sigma T ^ { 2 } = 4173.93 , \quad \Sigma Y T = 644.7 .$$
  2. Calculate the product moment correlation coefficient for these data.
  3. Comment on your result in relation to the student's enquiry.
Edexcel S1 2022 January Q1
11 marks Easy -1.2
  1. A factory produces shoes.
A quality control inspector at the factory checks a sample of 120 shoes for each of three types of defect. The Venn diagram represents the inspector's results. A represents the event that a shoe has defective stitching \(B\) represents the event that a shoe has defective colouring \(C\) represents the event that a shoe has defective soles \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-02_684_935_607_566} One of the shoes in the sample is selected at random.
  1. Find the probability that it does not have defective soles.
  2. Find \(\mathrm { P } \left( A \cap B \cap C ^ { \prime } \right)\)
  3. Find \(\mathrm { P } \left( A \cup B \cup C ^ { \prime } \right)\)
  4. Find the probability that the shoe has at most one type of defect.
  5. Given the selected shoe has at most one type of defect, find the probability it has defective stitching. The random variable \(X\) is the number of the events \(A , B , C\) that occur for a randomly selected shoe.
  6. Find \(\mathrm { E } ( X )\) \section*{This is a copy of the Venn diagram for this question.} \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-05_684_940_388_566}
Edexcel S1 2022 January Q2
6 marks Moderate -0.8
2. Tom's car holds 50 litres of petrol when the fuel tank is full. For each of 10 journeys, each starting with 50 litres of petrol in the fuel tank, Tom records the distance travelled, \(d\) kilometres, and the amount of petrol used, \(p\) litres. The summary statistics for the 10 journeys are given below. $$\sum d = 1029 \quad \sum p = 50.8 \quad \sum d p = 5240.8 \quad \mathrm {~S} _ { d d } = 344.9 \quad \mathrm {~S} _ { p p } = 0.576$$
  1. Calculate the product moment correlation coefficient between \(d\) and \(p\) The amount of petrol remaining in the fuel tank for each journey, \(w\) litres, is recorded.
    1. Write down an equation for \(w\) in terms of \(p\)
    2. Hence, write down the value of the product moment correlation coefficient between \(w\) and \(p\)
  3. Write down the value of the product moment correlation coefficient between \(d\) and \(w\)
Edexcel S1 2022 January Q3
10 marks Moderate -0.8
  1. The stem and leaf diagram shows the number of deliveries made by Pat each day for 24 days
\begin{table}[h]
\captionsetup{labelformat=empty} \caption{Key: 10 \(\mathbf { 8 }\) represents 108 deliveries}
1089(2)
1103666889999(11)
1245555558(8)
13\(a\)\(b\)\(c\)(3)
\end{table} where \(a\), \(b\) and \(c\) are positive integers with \(a < b < c\) An outlier is defined as any value greater than \(1.5 \times\) interquartile range above the upper quartile. Given that there is only one outlier for these data,
  1. show that \(c = 9\) The number of deliveries made by Pat each day is represented by \(d\) The data in the stem and leaf diagram are coded using $$x = d - 125$$ and the following summary statistics are obtained $$\sum x = - 96 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1306$$
  2. Find the mean number of deliveries.
  3. Find the standard deviation of the number of deliveries. One of these 24 days is selected at random. The random variable \(D\) represents the number of deliveries made by Pat on this day. The random variable \(X = D - 125\)
  4. Find \(\mathrm { P } ( D > 118 \mid X < 0 )\)
Edexcel S1 2022 January Q4
13 marks Moderate -0.3
  1. The random variable \(W\) has a discrete uniform distribution where
$$\mathrm { P } ( W = w ) = \frac { 1 } { 5 } \quad \text { for } w = 1,2,3,4,5$$
  1. Find \(\mathrm { P } ( 2 \leqslant W < 3.5 )\) The discrete random variable \(X = 5 - 2 W\)
  2. Find \(\mathrm { E } ( X )\)
  3. Find \(\mathrm { P } ( X < W )\) The discrete random variable \(\mathrm { Y } = \frac { 1 } { W }\)
  4. Find
    1. the probability distribution of \(Y\)
    2. \(\operatorname { Var } ( Y )\), showing your working.
  5. Find \(\operatorname { Var } ( 2 - 3 Y )\)
Edexcel S1 2022 January Q5
11 marks Standard +0.3
  1. Jia writes a computer program that randomly generates values from a normal distribution. He sets the mean as 40 and the standard deviation as 2.4
    1. Find the probability that a particular value generated by the computer program is less than 37
    Jia changes the mean to \(m\) but leaves the standard deviation as 2.4
    The computer program then randomly generates 2 independent values from this normal distribution. The probability that both of these values are greater than 32 is 0.16
  2. Find the value of \(m\), giving your answer to 2 decimal places. Jia now changes the mean to 4 and the standard deviation to 8
    The computer program then randomly generates 5 independent values from this normal distribution.
  3. Find the probability that at least one of these values is negative.
Edexcel S1 2022 January Q6
13 marks Moderate -0.8
  1. Students on a psychology course were given a pre-test at the start of the course and a final exam at the end of the course. The teacher recorded the number of marks achieved on the pre-test, \(p\), and the number of marks achieved on the final exam, \(f\), for 34 students and displayed them on the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-22_1121_1136_447_438}
The equation of the least squares regression line for these data is found to be $$f = 10.8 + 0.748 p$$ For these students, the mean number of marks on the pre-test is 62.4
  1. Use the regression model to find the mean number of marks on the final exam.
  2. Give an interpretation of the gradient of the regression line. Considering the equation of the regression line, Priya says that she would expect someone who scored 0 marks on the pre-test to score 10.8 marks on the final exam.
  3. Comment on the reliability of Priya's statement.
  4. Write down the number of marks achieved on the final exam for the student who exceeded the expectation of the regression model by the largest number of marks.
  5. Find the range of values of \(p\) for which this regression model, \(f = 10.8 + 0.748 p\), predicts a greater number of marks on the final exam than on the pre-test. Later the teacher discovers an error in the recorded data. The student who achieved a score of 98 on the pre-test, scored 92 not 29 on the final exam. The summary statistics used for the model \(f = 10.8 + 0.748 p\) are corrected to include this information and a new least squares regression line is found. Given the original summary statistics were, $$n = 34 \quad \sum p = 2120 \quad \sum p f = 133486 \quad \mathrm {~S} _ { p p } = 15573.76 \quad \mathrm {~S} _ { p f } = 11648.35$$
  6. calculate the gradient of the new regression line. Show your working clearly.