Questions — Edexcel S1 (606 questions)

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Edexcel S1 2008 June Q5
10 marks Easy -1.2
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
5 marks Moderate -0.8
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
12 marks Moderate -0.3
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
6 marks Easy -1.2
  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
11 marks Easy -1.2
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 marks Easy -1.3
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
13 marks Moderate -0.3
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
9 marks Moderate -0.8
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
15 marks Standard +0.3
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 marks Moderate -0.5
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
11 marks Moderate -0.3
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
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]{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
11 marks Easy -1.3
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
10 marks Moderate -0.8
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
14 marks Moderate -0.8
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
14 marks Moderate -0.8
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
12 marks Standard +0.3
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 2012 June Q1
10 marks Moderate -0.8
  1. A discrete random variable \(X\) has the probability function
$$\mathrm { P } ( X = x ) = \begin{cases} k ( 1 - x ) ^ { 2 } & x = - 1,0,1 \text { and } 2 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 1 } { 6 }\)
  2. Find \(\mathrm { E } ( X )\)
  3. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 4 } { 3 }\)
  4. Find \(\operatorname { Var } ( 1 - 3 X )\)
Edexcel S1 2012 June Q2
6 marks Moderate -0.8
2. A bank reviews its customer records at the end of each month to find out how many customers have become unemployed, \(u\), and how many have had their house repossessed, \(h\), during that month. The bank codes the data using variables \(x = \frac { u - 100 } { 3 }\) and \(y = \frac { h - 20 } { 7 }\) The results for the 12 months of 2009 are summarised below. $$\sum x = 477 \quad S _ { x x } = 5606.25 \quad \sum y = 480 \quad S _ { y y } = 4244 \quad \sum x y = 23070$$
  1. Calculate the value of the product moment correlation coefficient for \(x\) and \(y\).
  2. Write down the product moment correlation coefficient for \(u\) and \(h\). The bank claims that an increase in unemployment among its customers is associated with an increase in house repossessions.
  3. State, with a reason, whether or not the bank's claim is supported by these data.
Edexcel S1 2012 June Q3
15 marks Moderate -0.5
3. A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells. He collects a random sample of egg shells from each of 6 different nests and tests for pollutant level, \(p\), and measures the thinning of the shell, \(t\). The results are shown in the table below.
\(p\)3830251512
\(t\)1391056
[You may use \(\sum p ^ { 2 } = 1967\) and \(\sum p t = 694\) ]
  1. Draw a scatter diagram on the axes on page 7 to represent these data.
  2. Explain why a linear regression model may be appropriate to describe the relationship between \(p\) and \(t\).
  3. Calculate the value of \(S _ { p t }\) and the value of \(S _ { p p }\).
  4. Find the equation of the regression line of \(t\) on \(p\), giving your answer in the form \(t = a + b p\).
  5. Plot the point ( \(\bar { p } , \bar { t }\) ) and draw the regression line on your scatter diagram. The scientist reviews similar studies and finds that pollutant levels above 16 are likely to result in the death of a chick soon after hatching.
  6. Estimate the minimum thinning of the shell that is likely to result in the death of a chick. \includegraphics[max width=\textwidth, alt={}, center]{0593544d-392d-465b-b922-c9cb1435abb5-05_1257_1568_301_173}
Edexcel S1 2012 June Q4
9 marks Easy -1.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0593544d-392d-465b-b922-c9cb1435abb5-06_611_1127_237_447} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows how 25 people travelled to work.
Their travel to work is represented by the events $$\begin{array} { l l } B & \text { bicycle } \\ T & \text { train } \\ W & \text { walk } \end{array}$$
  1. Write down 2 of these events that are mutually exclusive. Give a reason for your answer.
  2. Determine whether or not \(B\) and \(T\) are independent events. One person is chosen at random.
    Find the probability that this person
  3. walks to work,
  4. travels to work by bicycle and train.
  5. Given that this person travels to work by bicycle, find the probability that they will also take the train.
Edexcel S1 2012 June Q5
13 marks Moderate -0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0593544d-392d-465b-b922-c9cb1435abb5-08_1031_1239_116_354} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A policeman records the speed of the traffic on a busy road with a 30 mph speed limit. He records the speeds of a sample of 450 cars. The histogram in Figure 2 represents the results.
  1. Calculate the number of cars that were exceeding the speed limit by at least 5 mph in the sample.
  2. Estimate the value of the mean speed of the cars in the sample.
  3. Estimate, to 1 decimal place, the value of the median speed of the cars in the sample.
  4. Comment on the shape of the distribution. Give a reason for your answer.
  5. State, with a reason, whether the estimate of the mean or the median is a better representation of the average speed of the traffic on the road.
Edexcel S1 2012 June Q6
10 marks Standard +0.3
  1. The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm .
    1. Find the probability that a randomly chosen adult female is taller than 150 cm .
      (3)
    Sarah is a young girl. She visits her doctor and is told that she is at the 60th percentile for height.
  2. Assuming that Sarah remains at the 60th percentile, estimate her height as an adult. The heights of an adult male population are normally distributed with standard deviation 9.0 cm . Given that \(90 \%\) of adult males are taller than the mean height of adult females,
  3. find the mean height of an adult male.
Edexcel S1 2012 June Q7
12 marks Moderate -0.8
  1. A manufacturer carried out a survey of the defects in their soft toys. It is found that the probability of a toy having poor stitching is 0.03 and that a toy with poor stitching has a probability of 0.7 of splitting open. A toy without poor stitching has a probability of 0.02 of splitting open.
    1. Draw a tree diagram to represent this information.
    2. Find the probability that a randomly chosen soft toy has exactly one of the two defects, poor stitching or splitting open.
      (3)
    The manufacturer also finds that soft toys can become faded with probability 0.05 and that this defect is independent of poor stitching or splitting open. A soft toy is chosen at random.
  2. Find the probability that the soft toy has none of these 3 defects.
  3. Find the probability that the soft toy has exactly one of these 3 defects.
Edexcel S1 2013 June Q1
10 marks Moderate -0.8
  1. Sammy is studying the number of units of gas, \(g\), and the number of units of electricity, \(e\), used in her house each week. A random sample of 10 weeks use was recorded and the data for each week were coded so that \(x = \frac { g - 60 } { 4 }\) and \(y = \frac { e } { 10 }\). The results for the coded data are summarised below
$$\sum x = 48.0 \quad \sum y = 58.0 \quad \mathrm {~S} _ { x x } = 312.1 \quad \mathrm {~S} _ { y y } = 2.10 \quad \mathrm {~S} _ { x y } = 18.35$$
  1. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\). Give the values of \(a\) and \(b\) correct to 3 significant figures.
  2. Hence find the equation of the regression line of \(e\) on \(g\) in the form \(e = c + d g\). Give the values of \(c\) and \(d\) correct to 2 significant figures.
  3. Use your regression equation to estimate the number of units of electricity used in a week when 100 units of gas were used.
  4. Find the probability distribution of \(X\) .
  5. Write down the value of \(\mathrm { F } ( 1.8 )\) .
  6. Find the probability distribution of \(X\) .勤