Questions — Edexcel S1 (606 questions)

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Edexcel S1 2013 January Q2
8 marks Moderate -0.8
2. The discrete random variable \(X\) can take only the values 1,2 and 3 . For these values the cumulative distribution function is defined by $$\mathrm { F } ( x ) = \frac { x ^ { 3 } + k } { 40 } \quad x = 1,2,3$$
  1. Show that \(k = 13\)
  2. Find the probability distribution of \(X\). Given that \(\operatorname { Var } ( X ) = \frac { 259 } { 320 }\)
  3. find the exact value of \(\operatorname { Var } ( 4 X - 5 )\).
Edexcel S1 2013 January Q3
10 marks Moderate -0.8
3. A biologist is comparing the intervals ( \(m\) seconds) between the mating calls of a certain species of tree frog and the surrounding temperature ( \(t { } ^ { \circ } \mathrm { C }\) ). The following results were obtained.
\(t { } ^ { \circ } \mathrm { C }\)813141515202530
\(m\) secs6.54.5654321
$$\text { (You may use } \sum t m = 469.5 , \quad \mathrm {~S} _ { t t } = 354 , \quad \mathrm {~S} _ { m m } = 25.5 \text { ) }$$
  1. Show that \(\mathrm { S } _ { t m } = - 90.5\)
  2. Find the equation of the regression line of \(m\) on \(t\) giving your answer in the form \(m = a + b t\).
  3. Use your regression line to estimate the time interval between mating calls when the surrounding temperature is \(10 ^ { \circ } \mathrm { C }\).
  4. Comment on the reliability of this estimate, giving a reason for your answer.
Edexcel S1 2013 January Q4
10 marks Standard +0.8
  1. The length of time, \(L\) hours, that a phone will work before it needs charging is normally distributed with a mean of 100 hours and a standard deviation of 15 hours.
    1. Find \(\mathrm { P } ( L > 127 )\).
    2. Find the value of \(d\) such that \(\mathrm { P } ( L < d ) = 0.10\)
    Alice is about to go on a 6 hour journey.
    Given that it is 127 hours since Alice last charged her phone,
  2. find the probability that her phone will not need charging before her journey is completed.
Edexcel S1 2013 January Q5
15 marks Moderate -0.8
  1. A survey of 100 households gave the following results for weekly income \(\pounds y\).
Income \(y\) (£)Mid-pointFrequency \(f\)
\(0 \leqslant y < 200\)10012
\(200 \leqslant y < 240\)22028
\(240 \leqslant y < 320\)28022
\(320 \leqslant y < 400\)36018
\(400 \leqslant y < 600\)50012
\(600 \leqslant y < 800\)7008
(You may use \(\sum f y ^ { 2 } = 12452\) 800)
A histogram was drawn and the class \(200 \leqslant y < 240\) was represented by a rectangle of width 2 cm and height 7 cm .
  1. Calculate the width and the height of the rectangle representing the class $$320 \leqslant y < 400$$
  2. Use linear interpolation to estimate the median weekly income to the nearest pound.
  3. Estimate the mean and the standard deviation of the weekly income for these data. One measure of skewness is \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\).
  4. Use this measure to calculate the skewness for these data and describe its value. Katie suggests using the random variable \(X\) which has a normal distribution with mean 320 and standard deviation 150 to model the weekly income for these data.
  5. Find \(\mathrm { P } ( 240 < X < 400 )\).
  6. With reference to your calculations in parts (d) and (e) and the data in the table, comment on Katie's suggestion.
Edexcel S1 2013 January Q6
13 marks Standard +0.3
6. A fair blue die has faces numbered \(1,1,3,3,5\) and 5 . The random variable \(B\) represents the score when the blue die is rolled.
  1. Write down the probability distribution for \(B\).
  2. State the name of this probability distribution.
  3. Write down the value of \(\mathrm { E } ( B )\). A second die is red and the random variable \(R\) represents the score when the red die is rolled. The probability distribution of \(R\) is
    \(r\)246
    \(\mathrm { P } ( R = r )\)\(\frac { 2 } { 3 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 6 }\)
  4. Find \(\mathrm { E } ( R )\).
  5. Find \(\operatorname { Var } ( R )\). Tom invites Avisha to play a game with these dice.
    Tom spins a fair coin with one side labelled 2 and the other side labelled 5 . When Avisha sees the number showing on the coin she then chooses one of the dice and rolls it. If the number showing on the die is greater than the number showing on the coin, Avisha wins, otherwise Tom wins. Avisha chooses the die which gives her the best chance of winning each time Tom spins the coin.
  6. Find the probability that Avisha wins the game, stating clearly which die she should use in each case.
Edexcel S1 2013 January Q7
12 marks Moderate -0.3
  1. Given that
$$\mathrm { P } ( A ) = 0.35 , \quad \mathrm { P } ( B ) = 0.45 \quad \text { and } \quad \mathrm { P } ( A \cap B ) = 0.13$$ find
  1. \(\mathrm { P } ( A \cup B )\)
  2. \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\) The event \(C\) has \(\mathrm { P } ( C ) = 0.20\) The events \(A\) and \(C\) are mutually exclusive and the events \(B\) and \(C\) are independent.
  3. Find \(\mathrm { P } ( B \cap C )\)
  4. Draw a Venn diagram to illustrate the events \(A , B\) and \(C\) and the probabilities for each region.
  5. Find \(\mathrm { P } \left( [ B \cup C ] ^ { \prime } \right)\)
Edexcel S1 2001 June Q1
5 marks Easy -1.2
  1. Each of the 25 students on a computer course recorded the number of minutes \(x\), to the nearest minute, spent surfing the internet during a given day. The results are summarised below.
$$\Sigma x = 1075 , \Sigma x ^ { 2 } = 44625 .$$
  1. Find \(\mu\) and \(\sigma\) for these data. Two other students surfed the internet on the same day for 35 and 51 minutes respectively.
  2. Without further calculation, explain the effect on the mean of including these two students.
    (2)
Edexcel S1 2001 June Q2
5 marks Easy -1.2
2. On a particular day in summer 1993 at 0800 hours the height above sea level, \(x\) metres, and the temperature, \(y ^ { \circ } \mathrm { C }\), were recorded in 10 Mediterranean towns. The following summary statistics were calculated from the results. $$\Sigma x = 7300 , \Sigma x ^ { 2 } = 6599600 , S _ { x y } = - 13060 , S _ { y y } = 140.9 .$$
  1. Find \(S _ { x x }\).
  2. Calculate, to 3 significant figures, the product moment correlation coefficient between \(x\) and \(y\).
  3. Give an interpretation of your coefficient.
Edexcel S1 2001 June Q3
8 marks Moderate -0.3
3. The continuous random variable \(Y\) is normally distributed with mean 100 and variance 256 .
  1. Find \(\mathrm { P } ( Y < 80 )\).
  2. Find \(k\) such that \(\mathrm { P } ( 100 - k \leq Y \leq 100 + k ) = 0.516\).
Edexcel S1 2001 June Q4
12 marks Easy -1.8
4. The discrete random variable \(X\) has the probability function shown in the table below.
\(x\)- 2- 10123
\(\mathrm { P } ( X = x )\)0.1\(\alpha\)0.30.20.10.1
Find
  1. \(\alpha\),
  2. \(\mathrm { P } ( - 1 < X \leq 2 )\),
  3. \(\mathrm { F } ( - 0.4 )\),
  4. \(\mathrm { E } ( 3 X + 4 )\),
  5. \(\operatorname { Var } ( 2 X + 3 )\).
Edexcel S1 2001 June Q5
13 marks Easy -1.2
5. A market researcher asked 100 adults which of the three newspapers \(A , B , C\) they read. The results showed that \(30 \operatorname { read } A , 26\) read \(B , 21\) read \(C , 5\) read both \(A\) and \(B , 7\) read both \(B\) and \(C , 6\) read both \(C\) and \(A\) and 2 read all three.
  1. Draw a Venn diagram to represent these data. One of the adults is then selected at random.
    Find the probability that she reads
  2. at least one of the newspapers,
  3. only \(A\),
  4. only one of the newspapers,
  5. \(A\) given that she reads only one newspaper.
Edexcel S1 2001 June Q6
16 marks Easy -1.2
6. Three swimmers Alan, Diane and Gopal record the number of lengths of the swimming pool they swim during each practice session over several weeks. The stem and leaf diagram below shows the results for Alan.
Lengths20 means 20
20122\(( 4 )\)
255667789\(( 7 )\)
3012224\(( 5 )\)
3566679\(( 5 )\)
401333333444\(( 10 )\)
45556667788999\(( 12 )\)
5000\(( 3 )\)
  1. Find the three quartiles for Alan's results. The table below summarises the results for Diane and Gopal.
    DianeGopal
    Smallest value3525
    Lower quartile3734
    Median4242
    Upper quartile5350
    Largest value6557
  2. Using the same scale and on the same sheet of graph paper draw box plots to represent the data for Alan, Diane and Gopal.
  3. Compare and contrast the three box plots.
Edexcel S1 2001 June Q7
16 marks Moderate -0.3
7. A music teacher monitored the sight-reading ability of one of her pupils over a 10 week period. At the end of each week, the pupil was given a new piece to sight-read and the teacher noted the number of errors \(y\). She also recorded the
number of hours \(x\) that the pupil had practised each week. The data are shown in the table below.
\(x\)1215711184693
\(y\)84138181215141216
  1. Plot these data on a scatter diagram.
  2. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\). $$\text { (You may use } \left. \Sigma x ^ { 2 } = 746 , \Sigma x y = 749 . \right)$$
  3. Give an interpretation of the slope and the intercept of your regression line.
  4. State whether or not you think the regression model is reasonable
    1. for the range of \(x\)-values given in the table,
    2. for all possible \(x\)-values. In each case justify your answer either by giving a reason for accepting the model or by suggesting an alternative model. END
Edexcel S1 2002 June Q1
4 marks Easy -1.8
  1. An unbiased die has faces numbered 1 to 6 inclusive. The die is rolled and the number that appears on the uppermost face is recorded.
    1. State the probability of not recording a 6 in one roll of the die.
    The die is thrown until a 6 is recorded.
  2. Find the probability that a 6 occurs for the first time on the third roll of the die.
    (3)
Edexcel S1 2002 June Q2
4 marks Easy -1.2
2. Statistical models can be used to describe real world problems. Explain the process involved in the formulation of a statistical model.
(4)
Edexcel S1 2002 June Q3
12 marks Moderate -0.8
3. For the events \(A\) and \(B\),
  1. explain in words the meaning of the term \(\mathrm { P } \left( \begin{array} { l l } B & A \end{array} \right)\),
  2. sketch a Venn diagram to illustrate the relationship \(\mathrm { P } \left( \begin{array} { l l } B & A \end{array} \right) = 0\). Three companies operate a bus service along a busy main road. Amber buses run \(50 \%\) of the service and \(2 \%\) of their buses are more than 5 minutes late. Blunder buses run \(30 \%\) of the service and \(10 \%\) of their buses are more than 5 minutes late. Clipper buses run the remainder of the service and only \(1 \%\) of their buses run more than 5 minutes late. Jean is waiting for a bus on the main road.
  3. Find the probability that the first bus to arrive is an Amber bus that is more than 5 minutes late. Let \(A , B\) and \(C\) denote the events that Jean catches an Amber bus, a Blunder bus and a Clipper bus respectively. Let \(L\) denote the event that Jean catches a bus that is more than 5 minutes late.
  4. Draw a Venn diagram to represent the events \(A , B , \mathrm { C }\) and \(L\). Calculate the probabilities associated with each region and write them in the appropriate places on the Venn diagram.
  5. Find the probability that Jean catches a bus that is more than 5 minutes late.
Edexcel S1 2002 June Q4
12 marks Moderate -0.8
4. A discrete random variable \(X\) takes only positive integer values. It has a cumulative distribution function \(\mathrm { F } ( x ) = \mathrm { P } ( X \leq x )\) defined in the table below.
\(X\)12345678
\(\mathrm {~F} ( x )\)0.10.20.250.40.50.60.751
  1. Determine the probability function, \(\mathrm { P } ( X = x )\), of \(X\).
  2. Calculate \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = 5.76\).
  3. Given that \(Y = 2 X + 3\), find the mean and variance of \(Y\).
Edexcel S1 2002 June Q5
12 marks Standard +0.3
5. A random variable \(X\) has a normal distribution.
  1. Describe two features of the distribution of \(X\). A company produces electronic components which have life spans that are normally distributed. Only \(1 \%\) of the components have a life span less than 3500 hours and \(2.5 \%\) have a life span greater than 5500 hours.
  2. Determine the mean and standard deviation of the life spans of the components. The company gives warranty of 4000 hours on the components.
  3. Find the proportion of components that the company can expect to replace under the warranty.
Edexcel S1 2002 June Q6
14 marks Moderate -0.3
6. The labelling on bags of garden compost indicates that the bags weigh 20 kg . The weights of a random sample of 50 bags are summarised in the table below.
Weight in kgFrequency
14.6-14.81
14.8-18.00
18.0-18.55
18.5-20.06
20.0-20.222
20.2-20.415
20.4-21.01
  1. On graph paper, draw a histogram of these data.
  2. Using the coding \(y = 10\) (weight in \(\mathrm { kg } - 14\) ), find an estimate for the mean and standard deviation of the weight of a bag of compost.
    [0pt] [Use \(\Sigma f y ^ { 2 } = 171\) 503.75]
  3. Using linear interpolation, estimate the median. The company that produces the bags of compost wants to improve the accuracy of the labelling. The company decides to put the average weight in kg on each bag.
  4. Write down which of these averages you would recommend the company to use. Give a reason for your answer.
Edexcel S1 2002 June Q7
16 marks Moderate -0.8
7. An ice cream seller believes that there is a relationship between the temperature on a summer day and the number of ice creams sold. Over a period of 10 days he records the temperature at 1 p.m., \(t ^ { \circ } \mathrm { C }\), and the number of ice creams sold, \(c\), in the next hour. The data he collects is summarised in the table below.
\(t\)\(c\)
1324
2255
1735
2045
1020
1530
1939
1219
1836
2354
[Use \(\left. \Sigma t ^ { 2 } = 3025 , \Sigma c ^ { 2 } = 14245 , \Sigma c t = 6526 .\right]\)
  1. Calculate the value of the product moment correlation coefficient between \(t\) and \(c\).
  2. State whether or not your value supports the use of a regression equation to predict the number of ice creams sold. Give a reason for your answer.
  3. Find the equation of the least squares regression line of \(c\) on \(t\) in the form \(c = a + b t\).
  4. Interpret the value of \(b\).
  5. Estimate the number of ice creams sold between 1 p.m. and 2 p.m. when the temperature at 1 p.m. is \(16 ^ { \circ } \mathrm { C }\).
    (3)
  6. At 1 p.m. on a particular day, the highest temperature for 50 years was recorded. Give a reason why you should not use the regression equation to predict ice cream sales on that day.
    (1)
Edexcel S1 2004 June Q1
5 marks Moderate -0.3
  1. A fair die has six faces numbered \(1,2,2,3,3\) and 3 . The die is rolled twice and the number showing on the uppermost face is recorded each time.
Find the probability that the sum of the two numbers recorded is at least 5 .
(5)
Edexcel S1 2004 June Q2
18 marks Moderate -0.8
2. A researcher thinks there is a link between a person's height and level of confidence. She measured the height \(h\), to the nearest cm , of a random sample of 9 people. She also devised a test to measure the level of confidence \(c\) of each person. The data are shown in the table below.
\(h\)179169187166162193161177168
\(c\)569561579561540598542565573
[You may use \(\Sigma h ^ { 2 } = 272094 , \Sigma c ^ { 2 } = 2878966 , \Sigma h c = 884484\) ]
  1. Draw a scatter diagram to illustrate these data.
  2. Find exact values of \(S _ { h c } S _ { h h }\) and \(S _ { c c }\).
  3. Calculate the value of the product moment correlation coefficient for these data.
  4. Give an interpretation of your correlation coefficient.
  5. Calculate the equation of the regression line of \(c\) on \(h\) in the form \(c = a + b h\).
  6. Estimate the level of confidence of a person of height 180 cm .
  7. State the range of values of \(h\) for which estimates of \(c\) are reliable.
Edexcel S1 2004 June Q3
13 marks Moderate -0.8
3. A discrete random variable \(X\) has a probability function as shown in the table below, where \(a\) and \(b\) are constants.
\(x\)0123
\(\mathrm { P } ( X = x )\)0.20.3\(b\)\(a\)
Given that \(\mathrm { E } ( X ) = 1.7\),
  1. find the value of \(a\) and the value of \(b\). Find
  2. \(\mathrm { P } ( 0 < X < 1.5 )\),
  3. \(\mathrm { E } ( 2 X - 3 )\).
  4. Show that \(\operatorname { Var } ( X ) = 1.41\).
  5. Evaluate \(\operatorname { Var } ( 2 X - 3 )\).
Edexcel S1 2004 June Q4
19 marks Easy -1.3
4. The attendance at college of a group of 18 students was recorded for a 4-week period. The number of students actually attending each of 16 classes are shown below.
18181717
16171618
18141718
15171816
    1. Calculate the mean and the standard deviation of the number of students attending these classes.
    2. Express the mean as a percentage of the 18 students in the group. In the same 4-week period, the attendance of a different group of 20, students is shown below.
      20161819
      15141415
      18151617
      16181514
  2. Construct a back-to-back stem and leaf diagram to represent the attendance in both groups.
  3. Find the mode, median and inter-quartile range for each group of students. The mean percentage attendance and standard deviation for the second group of students are 81.25 and 1.82 respectively.
  4. Compare and contrast the attendance of these 2 groups of students.
Edexcel S1 2004 June Q5
9 marks Standard +0.3
5. A health club lets members use, on each visit, its facilities for as long as they wish. The club's records suggest that the length of a visit can be modelled by a normal distribution with mean 90 minutes. Only \(20 \%\) of members stay for more than 125 minutes.
  1. Find the standard deviation of the normal distribution.
  2. Find the probability that a visit lasts less than 25 minutes. The club introduce a closing time of 10:00 pm. Tara arrives at the club at 8:00 pm.
  3. Explain whether or not this normal distribution is still a suitable model for the length of her visit.