Questions S1 (1967 questions)

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Edexcel S1 2012 January Q4
  1. The marks, \(x\), of 45 students randomly selected from those students who sat a mathematics examination are shown in the stem and leaf diagram below.
MarkTotals
36999\(( 3 )\)
40122234\(( 6 )\)
4566668\(( 5 )\)
50233344\(( 6 )\)
55566779\(( 6 )\)
600000013444\(( 9 )\)
65566789\(( 6 )\)
712333\(( 4 )\)
Key(3|6 means 36)
  1. Write down the modal mark of these students.
  2. Find the values of the lower quartile, the median and the upper quartile. For these students \(\sum x = 2497\) and \(\sum x ^ { 2 } = 143369\)
  3. Find the mean and the standard deviation of the marks of these students.
  4. Describe the skewness of the marks of these students, giving a reason for your answer. The mean and standard deviation of the marks of all the students who sat the examination were 55 and 10 respectively. The examiners decided that the total mark of each student should be scaled by subtracting 5 marks and then reducing the mark by a further \(10 \%\).
  5. Find the mean and standard deviation of the scaled marks of all the students.
Edexcel S1 2012 January Q5
  1. The age, \(t\) years, and weight, \(w\) grams, of each of 10 coins were recorded. These data are summarised below.
$$\sum t ^ { 2 } = 2688 \quad \sum t w = 1760.62 \quad \sum t = 158 \quad \sum w = 111.75 \quad S _ { w w } = 0.16$$
  1. Find \(S _ { t t }\) and \(S _ { t w }\) for these data.
  2. Calculate, to 3 significant figures, the product moment correlation coefficient between \(t\) and \(w\).
  3. Find the equation of the regression line of \(w\) on \(t\) in the form \(w = a + b t\)
  4. State, with a reason, which variable is the explanatory variable.
  5. Using this model, estimate
    1. the weight of a coin which is 5 years old,
    2. the effect of an increase of 4 years in age on the weight of a coin. It was discovered that a coin in the original sample, which was 5 years old and weighed 20 grams, was a fake.
  6. State, without any further calculations, whether the exclusion of this coin would increase or decrease the value of the product moment correlation coefficient. Give a reason for your answer.
Edexcel S1 2012 January Q6
  1. The following shows the results of a survey on the types of exercise taken by a group of 100 people.
65 run
48 swim
60 cycle
40 run and swim
30 swim and cycle
35 run and cycle
25 do all three
  1. Draw a Venn Diagram to represent these data. Find the probability that a randomly selected person from the survey
  2. takes none of these types of exercise,
  3. swims but does not run,
  4. takes at least two of these types of exercise. Jason is one of the above group.
    Given that Jason runs,
  5. find the probability that he swims but does not cycle.
Edexcel S1 2012 January Q7
  1. A manufacturer fills jars with coffee. The weight of coffee, \(W\) grams, in a jar can be modelled by a normal distribution with mean 232 grams and standard deviation 5 grams.
    1. Find \(\mathrm { P } ( W < 224 )\).
    2. Find the value of \(w\) such that \(\mathrm { P } ( 232 < W < w ) = 0.20\)
    Two jars of coffee are selected at random.
  2. Find the probability that only one of the jars contains between 232 grams and \(w\) grams of coffee.
Edexcel S1 2013 January Q1
  1. A teacher asked a random sample of 10 students to record the number of hours of television, \(t\), they watched in the week before their mock exam. She then calculated their grade, \(g\), in their mock exam. The results are summarised as follows.
$$\sum t = 258 \quad \sum t ^ { 2 } = 8702 \quad \sum g = 63.6 \quad \mathrm {~S} _ { g g } = 7.864 \quad \sum g t = 1550.2$$
  1. Find \(\mathrm { S } _ { t t }\) and \(\mathrm { S } _ { g t }\)
  2. Calculate, to 3 significant figures, the product moment correlation coefficient between \(t\) and \(g\). The teacher also recorded the number of hours of revision, \(v\), these 10 students completed during the week before their mock exam. The correlation coefficient between \(t\) and \(v\) was -0.753
  3. Describe, giving a reason, the nature of the correlation you would expect to find between \(v\) and \(g\).
Edexcel S1 2013 January Q2
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
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
  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
  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
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
  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
  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
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
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
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
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
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
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
  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
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
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
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
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
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
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)