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Year 13 Applied Testette 23/1 Name Question 1

Year 13 Applied Testette 23/1 Name Question 1 Intensity of light is measured in lumens. The random variable X represents the intensity of the light from a standard 100 watt light bulb. X is Normally distributed with mean 1720 and standard deviation 90. You may assume that the intensities for different bulbs are independent. (i) Show that P(X < 1700) = 0.4121. (4] (ii) These bulbs are sold in packs of 4. Find the probability that the intensities of exactly 2 of the 4 bulbs in a randomly chosen pack are below 1700 lumens. [3] A manufacturer claims that the average intensity of its 25 watt low energy light bulbs is 1720 lumens. A consumer organisation suspects that the true figure may be lower than this. The intensities of a random sample of 20 of these bulbs are measured. A hypothesis test is then carried out to check the claim. (iv) Write down a suitable null hypothesis and explain briefly why the alternative hypothesis should be H, : < 1720. State the meaning of y. (3] (v) Given that the standard deviation of the intensity of such bulbs is 90 lumens and that the mean intensity of the sample of 20 bulbs is 1703 lumens, carry out the test at the 5% significance level. {5] Two ships ? and Q are travelling at night with constant velocities. At midnight, P is at the point with position vector (20i + 10j) km relative to a fixed origin O. At the same time, Q is at the point with position vector (14i—6j)km. Three hours later, P is at the point with position vector (29i + 34j) km. The ship Q travels with velocity 12j kmh'. At time f¢ hours after midnight, the position vectors of Pand Q are pkm and qkm respectively. Find (a) the velocity of P, in terms of i and j, (2) (b) expressions for p and q, in terms of ¢, i and j. (4) At time ¢ hours after midnight, the distance between P and Q is dkm. => (c) By finding an expression for PQ, show that d?=25P—92t+292. (5) Weather conditions are such that an observer on P can only see the lights on Q when the distance between P and Q is 15 km or less. Given that when f= 1, the lights on Q move into sight of the observer, (d) find the time, to the nearest minute, at which the lights on Q move out of sight of the observer. (5)