Edexcel S1 — Question 8

Exam BoardEdexcel
ModuleS1 (Statistics 1)
TopicNormal Distribution
TypeExpected frequency with unknown parameter
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. Mathematical Formulae (Pink or Green) Nil \section*{Wednesday 13 January 2010 - Afternoon} Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S1), the paper reference (6683), your surname, other name and signature.
    Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2). There are 7 questions on this paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
    1. A jar contains 2 red, 1 blue and 1 green bead. Two beads are drawn at random from the jar without replacement.
    2. Draw a tree diagram to illustrate all the possible outcomes and associated probabilities. State your probabilities clearly.
    3. Find the probability that a blue bead and a green bead are drawn from the jar.
    4. The 19 employees of a company take an aptitude test. The scores out of 40 are illustrated in the stem and leaf diagram below.
    where \(a\) is a constant.
  5. Find the value of \(a\).
  6. Write down \(\mathrm { E } ( X )\).
  7. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 6 - 2 X\).
  8. Find \(\operatorname { Var } ( Y )\).
  9. Calculate \(\mathrm { P } ( X \geq Y )\). Shivani selects a ball and spins the appropriate coin.
  10. 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,
  11. find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
  12. Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.
    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]{e8afd947-55ac-424b-8db5-d5aa856ef4d7-064_264_615_360_443} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} One of these students is selected at random.
  13. Show that the probability that the student reads more than one magazine is \(\frac { 1 } { 6 }\).
  14. Find the probability that the student reads \(A\) or \(B\) (or both).
  15. Write down the probability that the student reads both \(A\) and \(C\). Given that the student reads at least one of the magazines,
  16. find the probability that the student reads \(C\).
  17. Determine whether or not reading magazine \(B\) and reading magazine \(C\) are statistically independent.
    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. where \(a , b\) and \(c\) are constants.
    The cumulative distribution function \(\mathrm { F } ( y )\) of \(Y\) is given in the following table.
  18. Estimate the number of motorists who were delayed between 8.5 and 13.5 minutes by the roadworks.
    (2)
    2. (a) State in words the relationship between two events \(R\) and \(S\) when \(\mathrm { P } ( R \cap S ) = 0\). The events \(A\) and \(B\) are independent with \(\mathrm { P } ( A ) = \frac { 1 } { 4 }\) and \(\mathrm { P } ( A \cup B ) = \frac { 2 } { 3 }\).
    Find
  19. \(\mathrm { P } ( B )\),
  20. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\),
  21. \(\mathrm { P } \left( B ^ { \prime } \mid A \right)\).
    3. The discrete random variable \(X\) can take only the values \(2,3,4\) or 6 . For these values the probability distribution function is given by [You may use \(\sum p ^ { 2 } = 1967\) and \(\sum p t = 694\) ]
  22. On graph paper, draw a scatter diagram to represent these data.
  23. Explain why a linear regression model may be appropriate to describe the relationship between \(p\) and \(t\).
  24. Calculate the value of \(S _ { p t }\) and the value of \(S _ { p p }\).
  25. Find the equation of the regression line of \(t\) on \(p\), giving your answer in the form \(t = a + b p\).
  26. 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.
  27. Estimate the minimum thinning of the shell that is likely to result in the death of a chick.
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e8afd947-55ac-424b-8db5-d5aa856ef4d7-077_401_741_296_1761} \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
    B bicycle
    \(T \quad\) train
    \(W\) walk
  28. Write down 2 of these events that are mutually exclusive. Give a reason for your answer.
  29. Determine whether or not \(B\) and \(T\) are independent events. One person is chosen at random.
    Find the probability that this person
  30. walks to work,
  31. travels to work by bicycle and train. Given that this person travels to work by bicycle,
  32. find the probability that they will also take the train.
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e8afd947-55ac-424b-8db5-d5aa856ef4d7-078_618_812_301_315} \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.
  33. Calculate the number of cars that were exceeding the speed limit by at least 5 mph in the sample.
  34. Estimate the value of the mean speed of the cars in the sample.
  35. Estimate, to 1 decimal place, the value of the median speed of the cars in the sample.
  36. Comment on the shape of the distribution. Give a reason for your answer.
  37. 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.
    6. The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm .
  38. 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.
  39. Assuming that Sarah remains at the 60th percentile, estimate her height as an adult.
    (3) 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,
  40. find the mean height of an adult male.
    (4)
    7. 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.
  41. Draw a tree diagram to represent this information.
    (3)
  42. 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.
  43. Find the probability that the soft toy has none of these 3 defects.
  44. Find the probability that the soft toy has exactly one of these 3 defects. \section*{Advanced Level} \section*{Friday 18 January 2013 - Afternoon} Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S1), the paper reference (6683), your surname, other name and signature.
    Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    This paper has 7 questions.
    The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner.
    Answers without working may not gain full credit. \section*{P41805A}
    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.
    $$\sum t = 258 \quad \sum t ^ { 2 } = 8702 \quad \sum g = 63.6 \quad \mathrm {~S} _ { g g } = 7.864 \quad \sum g t = 1550.2$$
  45. Find \(\mathrm { S } _ { t t }\) and \(\mathrm { S } _ { g t }\).
  46. 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 .
  47. Describe, giving a reason, the nature of the correlation you would expect to find between \(v\) and \(g\).
    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 $$\mathrm { F } ( x ) = \frac { x ^ { 3 } + k } { 40 } , \quad x = 1,2,3 .$$
  48. Show that \(k = 13\).
  49. Find the probability distribution of \(X\). Given that \(\operatorname { Var } ( X ) = \frac { 259 } { 320 }\),
  50. find the exact value of \(\operatorname { Var } ( 4 X - 5 )\).
    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. [You may assume that \(\sum h = 7150 , \sum t = 110 , \sum h ^ { 2 } = 7171500 , \sum t ^ { 2 } = 1716 , \sum t h = 64980\) and \(\mathrm { S } t t = 371.56\) ]
  51. Calculate \(\mathrm { S } _ { \text {th } }\) and \(\mathrm { S } _ { h h }\). Give your answers to 3 significant figures.
  52. Calculate the product moment correlation coefficient for this data.
  53. State whether or not your value supports the use of a regression equation to predict the air temperature at different heights on this mountain. Give a reason for your answer.
  54. Find the equation of the regression line of \(t\) on \(h\) giving your answer in the form \(t = a + b h\).
  55. Interpret the value of \(b\).
  56. Estimate the difference in air temperature between a height of 500 m and a height of 1000 m .
    2. The marks of a group of female students in a statistics test are summarised in Figure 1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e8afd947-55ac-424b-8db5-d5aa856ef4d7-083_412_723_331_390} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} An outlier is a mark that is
    either more than \(1.5 \times\) interquartile range above the upper quartile
    or more than \(1.5 \times\) interquartile range below the lower quartile.
  57. On graph paper draw a box plot to represent the marks of the male students, indicating clearly any outliers.
  58. Compare and contrast the marks of the male and the female students.
    (2) \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{(a) Write down the mark which is exceeded by \(75 \%\) of the female students.
    (1)
  59. Write down the mark which is exceeded by \(75 \%\) of the female students.
    The marks of a group of male students in the same statistics test are summarised by the stem and leaf diagram below.}
    Mark\(( 2 \mid 6\) means \(26 )\)Totals
    14\(( 1 )\)
    26\(( 1 )\)
    3447\(( 3 )\)
    4066778\(( 6 )\)
    5001113677\(( 9 )\)
    6223338\(( 6 )\)
    7008\(( 3 )\)
    85\(( 1 )\)
    90\(( 1 )\)
\end{table}
  • Find the median and interquartile range of the marks of the male students.
    3. In a company the 200 employees are classified as full-time workers, part-time workers or contractors. The table below shows the number of employees in each category and whether they walk to work or use some form of transport.
  • Find the probability distribution of \(X\).
  • Write down the value of \(\mathrm { F } ( 1.8 )\).
    3. An agriculturalist is studying the yields, \(y \mathrm {~kg}\), from tomato plants. The data from a random sample of 70 tomato plants are summarised below. [You may assume that \(\Sigma h = 7150 , \Sigma t = 110 , \Sigma h ^ { 2 } = 7171500 , \Sigma t ^ { 2 } = 1716 , \Sigma t h = 64980\) and \(\mathrm { S } t t = 371.56\) ]
  • Calculate \(\mathrm { S } _ { \text {th } }\) and \(\mathrm { S } _ { \text {hh } }\). Give your answers to 3 significant figures.
  • Calculate the product moment correlation coefficient for this data.
  • State whether or not your value supports the use of a regression equation to predict the air temperature at different heights on this mountain. Give a reason for your answer.
  • Find the equation of the regression line of \(t\) on \(h\) giving your answer in the form \(t = a + b h\).
  • Interpret the value of \(b\).
  • Estimate the difference in air temperature between a height of 500 m and a height of 1000 m .
    (2)
    2. The marks of a group of female students in a statistics test are summarised in Figure 1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e8afd947-55ac-424b-8db5-d5aa856ef4d7-090_417_732_328_1802} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
  • Write down the mark which is exceeded by \(75 \%\) of the female students. The marks of a group of male students in the same statistics test are summarised by the stem and leaf diagram below. (You may use \(\Sigma c = 111 , \Sigma c ^ { 2 } = 2375 , \Sigma s = 21 , \Sigma s ^ { 2 } = 79 , \Sigma c s = 380 , \mathrm {~S} _ { c c } = 321.5\).)
  • Calculate the value of \(\mathrm { S } _ { c s }\) and the value of \(\mathrm { S } _ { s s }\).
  • Calculate the product moment correlation coefficient for these data. Brad is not satisfied with his current internet service and decides to change his provider. He decides to pay a lot more for his new internet service.
  • On the basis of your calculation in part (b), comment on Brad's decision. Give a reason for your answer.
    2. A rugby club coach uses club records to take a random sample of 15 players from 1990 and an independent random sample of 15 players from 2010. The body weight of each player was recorded to the nearest kg and the results from 2010 are summarised in the table below. [You may use \(\Sigma x = 370 , \mathrm {~S} _ { x x } = 2587.5 , \Sigma y = 560 , \Sigma y ^ { 2 } = 39418 , \mathrm {~S} _ { x y } = - 710\) ]
  • Calculate \(\mathrm { S } _ { y y }\).
  • Calculate the product moment correlation coefficient for these data.
  • Interpret your value of the correlation coefficient. The researcher believes that a linear regression model may be appropriate to describe these data.
  • State, giving a reason, whether or not your value of the correlation coefficient supports the researcher's belief.
    (1)
  • Find the equation of the regression line of \(y\) on \(x\), giving your answer in the form \(y = a + b x\). Jack is a 40-year-old patient.
    1. Use your regression line to estimate the volume of blood pumped by each contraction of Jack's heart.
    2. Comment, giving a reason, on the reliability of your estimate.
      2. The table below shows the distances (to the nearest km ) travelled to work by the 50 employees in an office.
  • Show that \(p = 0.2\). Find
  • \(\mathrm { E } ( X )\)
  • \(\mathrm { F } ( 0 )\)
  • \(\mathrm { P } ( 3 X + 2 > 5 )\) Given that \(\operatorname { Var } ( X ) = 13.35\),
  • find the possible values of \(a\) such that \(\operatorname { Var } ( a X + 3 ) = 53.4\).
    2. The discrete random variable \(X\) has probability distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { 10 } \quad x = 1,2,3 , \ldots 10$$
  • Write down the name given to this distribution.
  • Write down the value of
    1. \(\mathrm { P } ( X = 10 )\)
    2. \(\mathrm { P } ( X < 10 )\) The continuous random variable \(Y\) has the normal distribution \(\mathrm { N } \left( 10,2 ^ { 2 } \right)\).
  • Write down the value of
    1. \(\mathrm { P } ( Y = 10 )\)
    2. \(\mathrm { P } ( Y < 10 )\)
      3. A large company is analysing how much money it spends on paper in its offices every year. The number of employees, \(x\), and the amount of money spent on paper, \(p\) ( \(\pounds\) hundreds), in 8 randomly selected offices are given in the table below. Key: 7|3| means 37 years for Greenslax and 31 years for Penville
      Some of the quartiles for these two distributions are given in the table below. You may use
      \(S _ { v v } = 42587.5\)
      \(S _ { v m } = 31512.5\)
      \(S _ { m m } = 25187.5\)
      \(\Sigma v = 19390\)
      \(\Sigma m = 10610\)
  • Find the product moment correlation coefficient between \(m\) and \(v\).
  • Give a reason to support fitting a regression model of the form \(m = a + b v\) to these data.
  • Find the value of \(b\) correct to 3 decimal places.
  • Find the equation of the regression line of \(m\) on \(v\).
  • Interpret your value of \(b\).
  • Use your answer to part (d) to estimate the amount of money spent when the number of visitors to the UK in a month is 2500000.
  • Comment on the reliability of your estimate in part (f). Give a reason for your answer.
    4. In a factory, three machines, \(J , K\) and \(L\), are used to make biscuits. Machine \(J\) makes \(25 \%\) of the biscuits.
    Machine \(K\) makes \(45 \%\) of the biscuits.
    The rest of the biscuits are made by machine \(L\).
    It is known that \(2 \%\) of the biscuits made by machine \(J\) are broken, \(3 \%\) of the biscuits made by machine \(K\) are broken and \(5 \%\) of the biscuits made by machine \(L\) are broken.
  • Draw a tree diagram to illustrate all the possible outcomes and associated probabilities. A biscuit is selected at random.
  • Calculate the probability that the biscuit is made by machine \(J\) and is not broken.
  • Calculate the probability that the biscuit is broken.
  • Given that the biscuit is broken, find the probability that it was not made by machine \(K\).
    5. The discrete random variable \(X\) has the probability function $$P ( X = x ) = \begin{cases} k x & x = 2,4,6
    k ( x - 2 ) x = 8 & x = 8
    0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  • Show that \(k = \frac { 1 } { 18 }\).
  • Find the exact value of \(\mathrm { F } ( 5 )\).
  • Find the exact value of \(\mathrm { E } ( X )\).
  • Find the exact value of \(\mathrm { E } \left( X ^ { 2 } \right)\).
  • Calculate \(\operatorname { Var } ( 3 - 4 X )\) giving your answer to 3 significant figures.
    6. The times, in seconds, spent in a queue at a supermarket by 85 randomly selected customers, are summarised in the table below.
    Time (seconds)Number of customers, \(f\)
    \(0 - 30\)2
    \(30 - 60\)10
    \(60 - 70\)17
    \(70 - 80\)25
    \(80 - 100\)25
    \(100 - 150\)6
    A histogram was drawn to represent these data. The \(30 - 60\) group was represented by a bar of width 1.5 cm and height 1 cm .
  • Find the width and the height of the \(70 - 80\) group.
  • Use linear interpolation to estimate the median of this distribution. Given that \(x\) denotes the midpoint of each group in the table and $$\Sigma f _ { x } = 6460 \quad \Sigma f _ { x ^ { 2 } } = 529400$$
  • calculate an estimate for
    1. the mean,
    2. the standard deviation,
      for the above data. One measure of skewness is given by $$\text { coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
  • Evaluate this coefficient and comment on the skewness of these data.
    7. The heights of adult females are normally distributed with mean 160 cm and standard deviation 8 cm .
  • Find the probability that a randomly selected adult female has a height greater than 170 cm . Any adult female whose height is greater than 170 cm is defined as tall.
    An adult female is chosen at random. Given that she is tall,
  • find the probability that she has a height greater than 180 cm . Half of tall adult females have a height greater than \(h \mathrm {~cm}\).
  • Find the value of \(h\).
    8. For the events \(A\) and \(B\), $$\mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.22 \text { and } \mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = 0.18$$
  • Find \(\mathrm { P } ( A )\).
  • Find \(\mathrm { P } ( A \cup B )\). Given that \(\mathrm { P } ( A \mid B ) = 0.6\),
  • find \(\mathrm { P } ( A \cap B )\).
  • Determine whether or not \(A\) and \(B\) are independent.
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