05(20A~1

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Nanyang Technological University *

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Nov 24, 2024

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From Examiner’s Report List of A level questions to attempt for 20 Aug Common Mistakes Lecture (Stats Edition) [Question numbers listed as in modified A level papers in Prelim Revision Package] S1 – 2015/P2/Q7, 2016/P2/Q6(iii), 2016/P2/Q10(ii) S2 – 2016/P2/Q12 S3 – 2015/P2/Q10 S4+S5 – 2015/P2/Q6, 2016/P2/Q9 S6 – 2013/P2/Q9, 2016/P2/Q11 Check if you made any of these common mistakes S1 - Assuming independence [       P P P A B A B    ] when not given OR not using it when you should - “Reclassifying” for P&C instead of using clearer cases - Not multiplying by no. of branches when taking       P P P A B C   S2 - Not recognizing binomial distribution - Using binomial distribution when p is not constant - Not checking that total probability = 1 S3 - Miscalculating variance [   Var X Y  ,   Var 2 X vs   1 2 Var X X  ] - Typing into GC 2  instead of  S4 - Using CLT to say ~ N X - Using CLT when not needed - Not using CLT when needed - Not using GC efficiently to find unbiased estimates S5 - Confusing 1 0 H :    or 1 0 H :    - Not defining  - Writing   2 ~ N , X x  , inconsistency with standardized form X Z n     - Using 2 s when 2  is given - Typing n  instead of  into Z-test - Not concluding in full with context S6 - Not labelling min and max x and y - Not keeping to regular intervals of x - Butter fingers keying wrong numbers in GC list / not excluding point as instructed - Not quoting enough s.f. for intermediate working - Using x on y when x is independent variable Hwa Chong Institution

S1 Modified 2015/P2/Q7 For events A , B and C it is given that P( ) 0.45 A  , P( ) 0.4 B  , P( ) 0.3 C  and P( ) 0.1 A B C    . It is also given that events A and B are independent, and that events A and C are independent. (i) Find P( | ) B A . [1] (ii) Given also that events B and C are independent, find P( ' ' ') A B C   . [3] (iii) Given instead that events B and C are not independent, find the greatest and least possible values of P( ' ' ') A B C   . [4] Modified 2016/P2/Q6(iii) In a game of chance, a player has to spin a fair spinner. The spinner has 7 sections and an arrow which has an equal chance of coming to rest over any of the 7 sections. The spinner has 1 section labelled R , 2 sections labelled B , and 4 sections labelled Y (see diagram). The player then has to throw one of three fair six-sided dice, coloured red, blue or yellow. If the spinner comes to rest over R the red die is thrown, if the spinner comes to rest over B the blue die is thrown and if the spinner comes to rest over Y the yellow die is thrown. The yellow die has one face with  on it, the blue die has two faces with  on it and the red die has three faces with  on it. The player wins the game if the die thrown comes to rest with a face showing  uppermost. (iii) Find the probability that a player wins 3 consecutive games, each time throwing a die of a different colour. [2] Modified 2016/P2/Q10(ii) The management board of a company consists of 6 men and 4 women. A chairperson, a secretary and a treasurer are chosen from the 10 members of the board. Find the number of ways the chairperson, the secretary and the treasurer can be chosen so that (ii) at least one is a woman and at least one is a man. [3] Hwa Chong Institution

S2 Modified 2016/P2/Q12 [SRJC00/P2/Q6 modified] A box contains 12 imported mangoes. 75% of them are found to be unripe. A fruit seller picks 4 mangoes randomly from the box. The random variable X is defined as the number of ripe mangoes picked. (i) Show that   P 0 X   14 55 , and find the probability distribution of X . Find E( ) X . [6] (ii) The ripe mangoes are sold at a cheaper price. For every pack of 4 mangoes, the fruit seller makes a profit of $2 if the pack contains more than 1 ripe mango; otherwise, he makes a profit of $3. Find his expected profit per pack. [3] (iii) It is also found that 1% of the mangoes he receives are rotten. A box of mangoes is considered substandard if it contains 2 or more rotten mangoes. Show that the probability that a randomly chosen box of mangoes is substandard is 00617 . 0 , correct to three significant figures. [2] (iv) The seller receives his goods by the lorry-load. A lorry-load consists of 50 randomly chosen boxes. Use a suitable distribution to estimate the probability that a lorry- load will include at most 2 boxes which are substandard. [3] S3 Modified 2015/P2/Q10 [2015 A Level/P2/Q12 modified] In this question you should state clearly the values of the parameters of any normal distribution you use. The masses in grams of apples have the distribution 2 N(300 , 20 ) and the masses in grams of pears have the distribution 2 N(200, 15 ) . A certain recipe required 5 apples and 8 pears. (i) Find the probability that the total mass of 5 randomly chosen apples is more than 1600 grams. [2] (ii) Find the probability that the total mass of 5 randomly chosen apples is more than the total mass of 8 randomly chosen pears. [3] The recipe requires the apples and pears to be prepared by peeling them and removing the cores. This process reduces the mass of each apple by 15% and the mass of each pear by 10%. (iii) Find the probability that the total mass, after preparation, of 5 randomly chosen apples and 8 randomly chosen pears is less than 2750 grams. [4] (iv) Find the probability that the mean mass, after preparation, of 5 randomly chosen apples is more than the mean mass, after preparation, of 8 randomly chosen pears by more than 95 grams. [4] Hwa Chong Institution

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S4+S5 Modified 2015/P2/Q6 [2015 A Level/P2/Q8 modified] A market stall sells pineapples which have masses that are normally distributed. The stall owner claims that the mean mass of the pineapples is at least 0.9 kg. Nur buys a random selection of 21 pineapples from the stall. The 21 pineapples have masses, in kg, as follows. Mass (kg) 0.80 0.81 0.82 0.85 0.89 0.90 0.91 0.93 0.96 1.00 No. of pineapples 2 1 4 2 3 2 3 2 1 1 Find unbiased estimates of the population mean and variance of the mass of pineapples. Test at the 3% level of significance whether there is any evidence to doubt the stall owner's claim. [7] Modified 2016/P2/Q9 [2016 A Level/P2/Q6 modified] The Managing Director of a company knows that, some years ago, the mean age of employees was 37 years. He believes that the mean age of employees now is less than 37 years. The Company Secretary obtains a suitable sample of 80 employees in order to carry out a hypothesis test of the Managing Director's belief that the mean age of the employees now is less than 37 years. You are given that the population variance of the ages is 140 years 2 . (i) Write down appropriate hypotheses to test the Managing Director's belief. You are given that the result of the test, using a 5% significance level, is that the Managing Director's belief should be accepted. Determine the set of possible values of the mean age of the sample of employees. [4] (ii) You are given instead that the mean age of the sample of employees is 35.2 years, and that the result of a test at the %  significance level is that the Managing Director's belief should not be accepted. Find the set of possible values of  . [3] S6 Modified 2013/P2/Q9 (i) Sketch a scatter diagram that might be expected when x and y are related approximately as given in each of the cases (A), (B) and (C) below. In each case your diagram should include 6 points, approximately equally spaced with respect to x , and with all x -values positive. The letters a , b , c , d , e and f represent constants. (A) 2 y a bx   , where a is positive and b is negative, (B) ln y c d x   , where c is positive and d is negative, (C) f y e x   , where e is positive and f is negative. [3] Hwa Chong Institution

A motoring website gives the following information about the distance travelled, y km, by a certain type of car at different speeds, x kmh -1 , on a fixed amount of fuel. (ii) Draw the scatter diagram for these values, labelling the axes. [1] (iii) Explain which of the three cases in part (i) is the most appropriate for modelling these values, and calculate the product moment correlation coefficient for this case. [2] (iv) It is required to estimate the distance travelled at a speed of 110 kmh -1 . Use the case that you identified in part (iii) to find the equation of a suitable regression line, and use your equation to find the required estimate. [3] Modified 2016/P2/Q11 A website about electric motors gives information about the percentage efficiency of motors depending on their power, measured in horsepower. Xian has copied the following table for a particular type of electric motor, but he has copied one of the efficiency values wrongly. Power, x 1 1.5 2 3 5 7.5 10 20 30 40 50 Efficiency, y % 72.5 82.5 84.0 87.4 87.5 88.5 89.5 90.2 91.0 91.7 92.4 (i) Plot a scatter diagram on graph paper for these values, labelling the axes, using a scale of 2cm to represent 10% efficiency on the y -axis and an appropriate scale for the x -axis. On your diagram, circle the point that Xian has copied wrongly. [2] For parts (ii) , (iii) and (iv) of this question you should exclude the point for which Xian has copied the efficiency value wrongly. (ii) Explain from your scatter diagram why the relationship between x and y should not be modelled by an equation of the form y ax b   . [1] (iii) Suppose that the relationship between x and y is modelled by an equation of the form c y d x   , where c and d are constants. State with a reason whether each of c and d is positive or negative. [2] (iv) Find the product moment correlation coefficient and the constants c and d for the model in part (iii) . [3] (v) Use the model c y d x   , with the values of c and d found in part (iv) , to estimate the efficiency value ( y ) that Xian has copied wrongly. Give two reasons why you would expect this estimate to be reliable. [3] Speed, x 88 96 104 112 120 128 Distance, y 148 147 144 138 126 107 Hwa Chong Institution