Questions S1 (1967 questions)

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Edexcel S1 2007 June Q7
7. The random variable \(X\) has probability distribution
\(x\)13579
\(\mathrm { P } ( X = x )\)0.2\(p\)0.2\(q\)0.15
  1. Given that \(\mathrm { E } ( X ) = 4.5\), write down two equations involving \(p\) and \(q\). Find
  2. the value of \(p\) and the value of \(q\),
  3. \(\mathrm { P } ( 4 < X \leqslant 7 )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 27.4\), find
  4. \(\operatorname { Var } ( X )\),
  5. \(\mathrm { E } ( 19 - 4 X )\),
  6. \(\operatorname { Var } ( 19 - 4 X )\).
Edexcel S1 2008 June Q1
  1. A disease is known to be present in \(2 \%\) of a population. A test is developed to help determine whether or not someone has the disease.
Given that a person has the disease, the test is positive with probability 0.95
Given that a person does not have the disease, the test is positive with probability 0.03
  1. Draw a tree diagram to represent this information. A person is selected at random from the population and tested for this disease.
  2. Find the probability that the test is positive. A doctor randomly selects a person from the population and tests him for the disease. Given that the test is positive,
  3. find the probability that he does not have the disease.
  4. Comment on the usefulness of this test.
Edexcel S1 2008 June Q2
2. The age in years of the residents of two hotels are shown in the back to back stem and leaf diagram below. Abbey Hotel \(8 | 5 | 0\) means 58 years in Abbey hotel and 50 years in Balmoral hotel Balmoral Hotel
(1)20
(4)97511
(4)983126(1)
(11)999976653323447(3)
(6)9877504005569(6)
\multirow[t]{3}{*}{(1)}85000013667(9)
6233457(6)
7015(3)
For the Balmoral Hotel,
  1. write down the mode of the age of the residents,
  2. find the values of the lower quartile, the median and the upper quartile.
    1. Find the mean, \(\bar { x }\), of the age of the residents.
    2. Given that \(\sum x ^ { 2 } = 81213\) find the standard deviation of the age of the residents. One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } }$$
  4. Evaluate this measure for the Balmoral Hotel. For the Abbey Hotel, the mode is 39 , the mean is 33.2 , the standard deviation is 12.7 and the measure of skewness is - 0.454
  5. Compare the two age distributions of the residents of each hotel.
Edexcel S1 2008 June Q3
3. The random variable \(X\) has probability distribution given in the table below.
\(x\)- 10123
\(\mathrm { P } ( X = x )\)\(p\)\(q\)0.20.150.15
Given that \(\mathrm { E } ( X ) = 0.55\), find
  1. the value of \(p\) and the value of \(q\),
  2. \(\operatorname { Var } ( X )\),
  3. \(\mathrm { E } ( 2 X - 4 )\).
Edexcel S1 2008 June Q4
4. Crickets make a noise. The pitch, \(v \mathrm { kHz }\), of the noise made by a cricket was recorded at 15 different temperatures, \(t ^ { \circ } \mathrm { C }\). These data are summarised below. $$\sum t ^ { 2 } = 10922.81 , \sum v ^ { 2 } = 42.3356 , \sum t v = 677.971 , \sum t = 401.3 , \sum v = 25.08$$
  1. Find \(S _ { t t } , S _ { v v }\) and \(S _ { t v }\) for these data.
  2. Find the product moment correlation coefficient between \(t\) and \(v\).
  3. State, with a reason, which variable is the explanatory variable.
  4. Give a reason to support fitting a regression model of the form \(v = a + b t\) to these data.
  5. Find the value of \(a\) and the value of \(b\). Give your answers to 3 significant figures.
  6. Using this model, predict the pitch of the noise at \(19 ^ { \circ } \mathrm { C }\).
Edexcel S1 2008 June Q5
5. A person's blood group is determined by whether or not it contains any of 3 substances \(A , B\) and \(C\). A doctor surveyed 300 patients' blood and produced the table below.
Blood containsNo. of Patients
only \(C\)100
\(A\) and \(C\) but not \(B\)100
only A30
\(B\) and \(C\) but not \(A\)25
only \(B\)12
\(A , B\) and \(C\)10
\(A\) and \(B\) but not \(C\)3
  1. Draw a Venn diagram to represent this information.
  2. Find the probability that a randomly chosen patient's blood contains substance \(C\). Harry is one of the patients. Given that his blood contains substance \(A\),
  3. find the probability that his blood contains all 3 substances. Patients whose blood contains none of these substances are called universal blood donors.
  4. Find the probability that a randomly chosen patient is a universal blood donor.
Edexcel S1 2008 June Q6
6. The discrete random variable \(X\) can take only the values 2,3 or 4 . For these values the cumulative distribution function is defined by $$F ( x ) = \frac { ( x + k ) ^ { 2 } } { 25 } \text { for } x = 2,3,4$$ where \(k\) is a positive integer.
  1. Find \(k\).
  2. Find the probability distribution of \(X\).
Edexcel S1 2008 June Q7
7. A packing plant fills bags with cement. The weight \(X \mathrm {~kg}\) of a bag of cement can be modelled by a normal distribution with mean 50 kg and standard deviation 2 kg .
  1. Find \(\mathrm { P } ( X > 53 )\).
  2. Find the weight that is exceeded by \(99 \%\) of the bags. Three bags are selected at random.
  3. Find the probability that two weigh more than 53 kg and one weighs less than 53 kg .
Edexcel S1 2009 June Q1
  1. The volume of a sample of gas is kept constant. The gas is heated and the pressure, \(p\), is measured at 10 different temperatures, \(t\). The results are summarised below.
    \(\sum p = 445 \quad \sum p ^ { 2 } = 38125 \quad \sum t = 240 \quad \sum t ^ { 2 } = 27520 \quad \sum p t = 26830\)
    1. Find \(\mathrm { S } _ { p p }\) and \(\mathrm { S } _ { p t }\).
    Given that \(\mathrm { S } _ { t t } = 21760\),
  2. calculate the product moment correlation coefficient.
  3. Give an interpretation of your answer to part (b).
Edexcel S1 2009 June Q2
2. On a randomly chosen day the probability that Bill travels to school by car, by bicycle or on foot is \(\frac { 1 } { 2 } , \frac { 1 } { 6 }\) and \(\frac { 1 } { 3 }\) respectively. The probability of being late when using these methods of travel is \(\frac { 1 } { 5 } , \frac { 2 } { 5 }\) and \(\frac { 1 } { 10 }\) respectively.
  1. Draw a tree diagram to represent this information.
  2. Find the probability that on a randomly chosen day
    1. Bill travels by foot and is late,
    2. Bill is not late.
  3. Given that Bill is late, find the probability that he did not travel on foot.
Edexcel S1 2009 June Q3
3. The variable \(x\) was measured to the nearest whole number. Forty observations are given in the table below.
\(x\)\(10 - 15\)\(16 - 18\)\(19 -\)
Frequency15916
A histogram was drawn and the bar representing the \(10 - 15\) class has a width of 2 cm and a height of 5 cm . For the \(16 - 18\) class find
  1. the width,
  2. the height
    of the bar representing this class.
Edexcel S1 2009 June Q4
4. A researcher measured the foot lengths of a random sample of 120 ten-year-old children. The lengths are summarised in the table below.
Foot length, \(l\), (cm)Number of children
\(10 \leqslant l < 12\)5
\(12 \leqslant l < 17\)53
\(17 \leqslant l < 19\)29
\(19 \leqslant l < 21\)15
\(21 \leqslant l < 23\)11
\(23 \leqslant l < 25\)7
  1. Use interpolation to estimate the median of this distribution.
  2. Calculate estimates for the mean and the standard deviation of these data. One measure of skewness is given by $$\text { Coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
  3. Evaluate this coefficient and comment on the skewness of these data. Greg suggests that a normal distribution is a suitable model for the foot lengths of ten-year-old children.
  4. Using the value found in part (c), comment on Greg's suggestion, giving a reason for your answer.
Edexcel S1 2009 June Q5
5. The weight, \(w\) grams, and the length, \(l \mathrm {~mm}\), of 10 randomly selected newborn turtles are given in the table below.
\(l\)49.052.053.054.554.153.450.051.649.551.2
\(w\)29323439383530312930
$$\text { (You may use } \mathrm { S } _ { l l } = 33.381 \quad \mathrm {~S} _ { w l } = 59.99 \quad \mathrm {~S} _ { w w } = 120.1 \text { ) }$$
  1. Find the equation of the regression line of \(w\) on \(l\) in the form \(w = a + b l\).
  2. Use your regression line to estimate the weight of a newborn turtle of length 60 mm .
  3. Comment on the reliability of your estimate giving a reason for your answer.
Edexcel S1 2009 June Q6
6. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } a ( 3 - x ) & x = 0,1,2
b & x = 3 \end{array} \right.$$
  1. Find \(\mathrm { P } ( X = 2 )\) and complete the table below.
    \(x\)0123
    \(\mathrm { P } ( X = x )\)\(3 a\)\(2 a\)\(b\)
    Given that \(\mathrm { E } ( X ) = 1.6\)
  2. Find the value of \(a\) and the value of \(b\). Find
  3. \(\mathrm { P } ( 0.5 < X < 3 )\),
  4. \(\mathrm { E } ( 3 X - 2 )\).
  5. Show that the \(\operatorname { Var } ( X ) = 1.64\)
  6. Calculate \(\operatorname { Var } ( 3 X - 2 )\).
Edexcel S1 2009 June Q7
7. (a) Given that \(\mathrm { P } ( A ) = a\) and \(\mathrm { P } ( B ) = b\) express \(\mathrm { P } ( A \cup B )\) in terms of \(a\) and \(b\) when
  1. \(A\) and \(B\) are mutually exclusive,
  2. \(A\) and \(B\) are independent. Two events \(R\) and \(Q\) are such that
    \(\mathrm { P } \left( R \cap Q ^ { \prime } \right) = 0.15 , \quad \mathrm { P } ( Q ) = 0.35\) and \(\mathrm { P } ( R \mid Q ) = 0.1\)
    Find the value of
    (b) \(\mathrm { P } ( R \cup Q )\),
    (c) \(\mathrm { P } ( R \cap Q )\),
    (d) \(\mathrm { P } ( R )\).
Edexcel S1 2009 June Q8
8. The lifetimes of bulbs used in a lamp are normally distributed. A company \(X\) sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.
  1. Find the probability of a bulb, from company \(X\), having a lifetime of less than 830 hours.
  2. In a box of 500 bulbs, from company \(X\), find the expected number having a lifetime of less than 830 hours. A rival company \(Y\) sells bulbs with a mean lifetime of 860 hours and \(20 \%\) of these bulbs have a lifetime of less than 818 hours.
  3. Find the standard deviation of the lifetimes of bulbs from company \(Y\). Both companies sell the bulbs for the same price.
  4. State which company you would recommend. Give reasons for your answer.
Edexcel S1 2010 June Q2
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]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_787_395_734_548} \section*{Coin}
    \includegraphics[max width=\textwidth, alt={}]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_1007_488_808_950}
    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 2010 June Q3
3. 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 2010 June Q4
4. 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]
\includegraphics[alt={},max width=\textwidth]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-07_397_934_374_502} \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 2010 June Q5
5. 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 2010 June Q6
6. 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]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-11_1013_1701_1718_116}
Edexcel S1 2010 June Q7
7. 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 $$h = Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \quad \text { and } \quad k = Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
  4. Find the value of \(h\) and the value of \(k\). An employee is selected at random.
  5. Find the probability that the distance travelled to work by this employee is an outlier.
Edexcel S1 2011 June Q1
  1. On a particular day the height above sea level, \(x\) metres, and the mid-day temperature, \(y ^ { \circ } \mathrm { C }\), were recorded in 8 north European towns. These data are summarised below
$$\mathrm { S } _ { x x } = 3535237.5 \quad \sum y = 181 \quad \sum y ^ { 2 } = 4305 \quad \mathrm {~S} _ { x y } = - 23726.25$$
  1. Find \(\mathrm { S } _ { y y }\)
  2. Calculate, to 3 significant figures, the product moment correlation coefficient for these data.
  3. Give an interpretation of your coefficient. A student thought that the calculations would be simpler if the height above sea level, \(h\), was measured in kilometres and used the variable \(h = \frac { x } { 1000 }\) instead of \(x\).
  4. Write down the value of \(\mathrm { S } _ { h h }\)
  5. Write down the value of the correlation coefficient between \(h\) and \(y\).
Edexcel S1 2011 June Q2
  1. The random variable \(X \sim \mathrm {~N} \left( \mu , 5 ^ { 2 } \right)\) and \(\mathrm { P } ( X < 23 ) = 0.9192\)
    1. Find the value of \(\mu\).
    2. Write down the value of \(\mathrm { P } ( \mu < X < 23 )\).
    3. The discrete random variable \(Y\) has probability distribution
    \(y\)1234
    \(\mathrm { P } ( Y = y )\)\(a\)\(b\)0.3\(c\)
    where \(a , b\) and \(c\) are constants. The cumulative distribution function \(\mathrm { F } ( y )\) of \(Y\) is given in the following table
    \(y\)1234
    \(\mathrm {~F} ( y )\)0.10.5\(d\)1.0
    where \(d\) is a constant.
  2. Find the value of \(a\), the value of \(b\), the value of \(c\) and the value of \(d\).
  3. Find \(\mathrm { P } ( 3 Y + 2 \geqslant 8 )\).
Edexcel S1 2011 June Q4
4. Past records show that the times, in seconds, taken to run 100 m by children at a school can be modelled by a normal distribution with a mean of 16.12 and a standard deviation of 1.60 A child from the school is selected at random.
  1. Find the probability that this child runs 100 m in less than 15 s . On sports day the school awards certificates to the fastest \(30 \%\) of the children in the 100 m race.
  2. Estimate, to 2 decimal places, the slowest time taken to run 100 m for which a child will be awarded a certificate.