Edexcel S1 (Statistics 1) 2019 January

1. The Venn diagram shows the probability of a randomly selected student from a school being in the sets \(L , B\) and \(C\), where
   \(L\) represents the event that the student has instrumental music lessons
   \(B\) represents the event that the student plays in the school band
   \(C\) represents the event that the student sings in the school choir
   \(p , q , r\) and \(s\) are probabilities.
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   1. Select a pair of mutually exclusive events from \(L , B\) and \(C\).
   Given that \(\mathrm { P } ( L ) = 0.4 , \mathrm { P } ( B ) = 0.13 , \mathrm { P } ( C ) = 0.3\) and the events \(L\) and \(C\) are independent,
2. find the value of \(p\),
3. find the value of \(q\), the value of \(r\) and the value of \(s\). A student is selected at random from those who play in the school band or sing in the school choir.
4. Find the exact probability that this student has instrumental music lessons.

2. The discrete random variable \(X\) has the following probability distribution.

1. Find \(\mathrm { E } ( X )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 4.54\)
2. find the value of \(a\) and the value of \(b\). The random variable \(Y = 3 - 2 X\)
3. Find \(\operatorname { Var } ( Y )\).

1. The weights of women boxers in a tournament are normally distributed with mean 64 kg and standard deviation 8 kg .
   1. Find the probability that a randomly chosen woman boxer in the tournament weighs less than 51 kg .
   In the tournament, women boxers who weigh less than 51 kg are classified as lightweight. Ren weighs 49 kg and she has a match against another randomly selected, lightweight woman boxer.
2. Find the probability that Ren weighs less than the other boxer. In the tournament, women boxers who weigh more than \(H \mathrm {~kg}\) are classified as heavyweight. Given that \(10 \%\) of the women boxers in the tournament are classified as heavyweight,
3. find the value of \(H\).

4. A group of 100 adults recorded the amount of time, \(t\) minutes, they spent exercising each day. Their results are summarised in the table below. [You may use \(\sum \mathrm { f } x ^ { 2 } = 455\) 512.5]
A histogram is drawn to represent these data.
The bar representing the time \(0 \leqslant t < 15\) has width 0.5 cm and height 6 cm .

1. Calculate the width and height of the bar representing a time of \(60 \leqslant t < 120\)
2. Use linear interpolation to estimate the median time spent exercising by these adults each day.
3. Find an estimate of the mean time spent exercising by these adults each day.
4. Calculate an estimate for the standard deviation of these times.
5. Describe, giving a reason, the skewness of these data. Further analysis of the above data revealed that 18 of the 25 adults in the \(0 \leqslant t < 15\) group took no exercise each day.
6. State, giving a reason, what effect, if any, this new information would have on your answers to
   1. the estimate of the median in part (b),
   2. the estimate of the mean in part (c),
   3. the estimate of the standard deviation in part (d).

1. Some children are playing a game involving throwing a ball into a bucket. Each child has 3 throws and the number of times the ball lands in the bucket, \(x\), is recorded. Their results are given in the table below.

1. Find \(\bar { x }\)
   (1) Sandra decides to model the game by assuming that on each throw, the probability of the ball landing in the bucket is 0.4 for every child on every throw and that the throws are all independent. The random variable \(S\) represents the number of times the ball lands in the bucket for a randomly selected child.
2. Find \(\mathrm { P } ( S = 2 )\)
3. Complete the table below to show the probability distribution for \(S\).

   \(s\)	0	1	2	3
   \(\mathrm { P } ( S = s )\)		0.432		0.064

   Ting believes that the probability of the ball landing in the bucket is not the same for each throw. He suggests that the probability will increase with each throw and uses the model $$p _ { i } = 0.15 i + 0.10$$ where \(i = 1,2,3\) and \(p _ { i }\) is the probability that the \(i\) th throw of the ball, by any particular child, will land in the bucket.
   The random variable \(T\) represents the number of times the ball lands in the bucket for a randomly selected child using Ting’s model.
4. Show that
   1. \(\mathrm { P } ( T = 3 ) = 0.055\)
   2. \(\mathrm { P } ( T = 1 ) = 0.45\)
      (5)
5. Complete the table below to show the probability distribution for \(T\), stating the exact probabilities in each case.

   \(t\)	0	1	2	3
   \(\mathrm { P } ( T = t )\)		0.45		0.055

6. State, giving your reasons, whether Sandra's model or Ting's model is the more appropriate for modelling this game.

1. Following some school examinations, Chetna is studying the results of the 16 students in her class. The mark for paper \(1 , x\), and the mark for paper \(2 , y\), for each student are summarised in the following statistics.

$$\bar { x } = 35.75 \quad \bar { y } = 25.75 \quad \sigma _ { x } = 7.79 \quad \sigma _ { y } = 11.91 \quad \sum x y = 15837$$

1. Comment on the differences between the marks of the students on paper 1 and paper 2 Chetna decides to examine these data in more detail and plots the marks for each of the 16 students on the scatter diagram opposite.
2. 1. Explain why the circled point \(( 38,0 )\) is possibly an outlier.
   2. Suggest a possible reason for this result. Chetna decides to omit the data point \(( 38,0 )\) and examine the other 15 students' marks.
3. Find the value of \(\bar { x }\) and the value of \(\bar { y }\) for these 15 students. For these 15 students
4. 1. explain why \(\sum x y\) is still 15837
   2. show that \(\mathrm { S } _ { x y } = 1169.8\) For these 15 students, Chetna calculates \(\mathrm { S } _ { x x } = 965.6\) and \(\mathrm { S } _ { y y } = 1561.7\) correct to 1 decimal place.
5. Calculate the product moment correlation coefficient for these 15 students.
6. Calculate the equation of the line of regression of \(y\) on \(x\) for these 15 students, giving your answer in the form \(y = a + b x\) The product moment correlation coefficient between \(x\) and \(y\) for all 16 students is 0.746
7. Explain how your calculation in part (e) supports Chetna's decision to omit the point \(( 38,0 )\) before calculating the equation of the linear regression line.
   (1)
8. Estimate the mark in the second paper for a student who scored 38 marks in the first paper.
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