Section A [45 marks] marks] Answer all questions in this section
1. In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, y minutes, for the 11 students were 11, 23, 35, 36, 51, 53, 54, 55, 60, 77, 110. (a) Find the median and quartiles for these data. [3 marks] marks] (b) Determine if there is any outlier. [2 marks] marks] (c) Hence draw a box-plot to represent the points above. [2 marks] marks]
2. A box contains 15 tulip bulbs of which 10 will have red flowers and 5 yellow, though it is not possible to determine the colour of the flowers by visual inspection of the bulbs. A package of 5 is to be made up from the 15. If they are selected at random, what is the probability that the package will contain exactly 3 red-flowering bulbs? [3 marks] marks] (a) One of the 15 bulbs is found to be damaged and is discarded before the selection is made. Find the conditional probability that the package of 5 contains exactly 3 red-flowering bulbs, given that the damaged damaged bulb is red. [2 marks] marks] (b) Hence or otherwise, find the probability that the package of 5 contains exactly 3 red-flowering bulbs when one bulb in the original 15 has been thrown away as damaged, given that all the bulbs are equally likely to be damaged. damaged. [3 marks] marks]
3. Anne plays a game in which a fair six-sided die is thrown once. If the score is 1, 2 or 3, Anne loses RM10. If the score is 4 or 5, Anne wins RM x RM x.. If the score is 6, Anne wins RM 2x. 2x. (a) Tabulate Anne profit in distribution table. [3 marks] marks] 2
(b) Show that the expectation of Anne’s profit i s RM( RM( x 5 ) in a single game.
[2 marks] marks]
(c) Calculate the value of x of x for which, on average, Anne’s profit is zero. (d) Given that x that x = 12, calculate the variance of Anne’s profit in a single game.
[1 mark ] [4 marks] marks]
3
4. Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below. Candidate Manager X Manager X Manager Y
A 62 54
B 56 47
C 87 71
D 54 50
E 65 49
F 15 25
G 12 30
H 10 44
(a) Calculate the Spearman’s rank correlation coefficient for these data and data and comment on the result. [5 marks] marks] (b) Manager Y later later discovered he had miscopied his score for candidate D candidate D and it should be 54. Without carrying out any further calculations, explain how you would calculate Spearman’s rank correlation in this case. [1 mark ] 1|5
5. Calculate the simple aggregate aggregate price index for year 2011 for three types of houses X, houses X, Y and Z and Z with with 2008 as the base year: Price ( RM ‘000 )
Type of houses X Y Z
2008
2011
200 450 750
800 860 1200
Comment Comment on the value that you obtained.
[5 marks] marks]
6. A certain country wants to find out the unemployment rate in the country. The table below shows the number of unemployed people in the different quarters from the year 2010 to 2012. Number of unemployed ( Thousands ) Year 2010 2011 2012
Quarter 1
2
3
4
36 32 28
24 23 21
25 23 20
30 27 25
The seasonal variation based on the additive model are as follows: Quarter 1 +4
2 -3
3 -3
4 +2
(a) Write the equation of the trend line. (b) Forecast the number of unemployed unemployed people for the first two quarters of 2013.
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[5 marks] marks] [4 marks] marks]
Section B (15 marks)
Answer any one question in this section
7. In a large population the proportion having blood group A is 35%. Specimens of blood from the first five people attending a clinic are to be tested. It can be assumed that these five people are a random sample from the population. The random variable X variable X denotes denotes the number of people in the sample who are found to have blood group A. (a) Show that P( X 2 ) = 0.765, correct to 3 decimal decimal places. [3 marks] marks] Three such samples of five people are taken. Find (b) the probability that each of these three samples has more than two people with blood group A. [3 marks] marks] (c) the probability that one of these three samples has exactly one person with blood group A, another has exactly two person with blood group A, and the remaining sample has more than two people with blood group A. [3 marks] marks] One hundred such samples of five people are taken. Using a suitable approximation, find the probability that the number of these samples that contain two or fewer people with blood group A will be at least 70. [6 marks] marks]
8. A car manufacturer is testing the braking distance for a new model of car. The table shows the braking distance, y distance, y metres, for different speeds, x speeds, x km/h, km/h, when the brakes were applied. Car speed, x speed, x km/h Braking distance, y distance, y metres
30 25
50 50
70 85
90 155
110 235
130 350
(a) Plot a scatter diagram. [2 marks] marks] (b) Calculate the equation of the regression line of y of y on on x x and and draw the line l ine on the scatter diagram. [4 marks] marks] (c) Use your regression equation to predict the values of y when y when x x = 100 and x = x = 150. Comment, with reason, on the likely accuracy accuracy of these predictions. [2 marks] marks] If we assume that the data can be modeled by y
ax b ,
(d) determine the value of a and b. [4 marks] marks] (e) Hence predict the values of y when y when x x = = 100 and x and x = = 150 and give your comment on the accuracy of the predictions. [3 marks] marks]
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