RI_8865_2022_Prelim Stats

___________________________________________________________________________ H1 Mathematics 2022 Year 6 Preliminary Examination Page 1 of 3 Section B: Statistics [60 marks] 6 A bus runs at regular intervals. The delay time, X, is the number of minutes that the bus arrived after its scheduled arrival time at the interchange. It is given that X follows a normal distribution, and that a bus arrives earl ier than the scheduled arrival time at the interchange with a probability of 0 .06 and at least 15 minutes late with a probability of 0.029. (i) Find the mean and standard deviation of X. [4] (ii) Sketch this distribution for delay times between −8 and 8 minutes. [2] 7 A and B are events such that P (A) = 1 5 , P (A  B) = 9 20 , P (AB) = 1 6 and P (A  B) = k, where 0  k  1. (i) Find the value of k. [3] (ii) Find P (A'  B). [2] (iii) Explain whether events A' and B are independent. [1] 8 A factory makes lamps. The probability that a lamp is defective is 0.05. A random sample of 30 lamps is tested. (i) Find the expected number of defective lamps in the sample. [1] (ii) Find the probability that there is at least one defective lamp in the sample. [2] (iii) Given that there is at least one defective lamp in the sample, find the probability that there are at most two defective lamps in the sample. [3] 9 Nine prosecutors were asked the number of years, x (years), of experience each of them had as a prosecutor, and the percentage, y (%), of their cases that ended in guilty pleas. The data obtained was recorded below: x 10 12 8 1 3 2 7 15 5 y 93 90 82 72 76 70 84 95 83 (i) Draw a scatter diagram for the above data. [2] (ii) Find the value of r, the product moment correlation coefficient, and comment on its value in the context of the question. [2] (iii) Find the equation of the regression line of y on x, giving your answer in the form y = mx + c, with the values of m and c correct to 3 significant figures. Interpret the value of m in this context. [2]

___________________________________________________________________________ ___________________________________________________________________________ H1 Mathematics 2022 Year 6 Preliminary Examination Page 2 of 3 (iv) Use the equation of your regression line to calculate an estimate for the percentage of cases ending in guilty pleas for a prosecutor with 20 years of experience. Comment on the reliability of your estimate. [3] 10 In this question you should state the parameters of any distributions that you use. A fast-food chain sells fries in both small and large packets. The mass of fries (in grams) in a small packet follows the normal distribution N (80, 2 2) and the mass of fries (in grams) in a large packet follows the normal distribution N (150, 3 2). The distributions are independent. (i) Two small packets of fries are randomly chosen. Find the probability that