<p> Department of Mathematics</p><p>Center for Foundation Studies, IIUM</p><p>Semester I, 2013/2014</p><p>SHF1124 (MATHEMATICS II)</p><p>TUTORIAL 6</p><p>CHAPTER 6: THE NORMAL DISTRIBUTION</p><p>Page 1 of 10 Page</p><p>Section in Textbook Questions</p><p>Number</p><p>8.2 The Standard Normal Distribution 360-361 13, 14</p><p>8.3 Applications of the Normal 366-367 3, 4, 5, 7</p><p>Distribution</p><p>8.4 Central Limit Theorem 383-384 6, 12</p><p>8.5 The Normal Approximation to the 391 5, 7, 8, 11</p><p>Binomial and Poisson Distribution</p><p>Review Exercises 394-396 6, 8, 10 (a, b, c), 12, 13</p><p>*Required Textbook: - Salina Mohin et al , Mathematics & Statistics for Pre-University,</p><p>Page 2 of 10 McGraw-Hill Education (Malaysia) Sdn Bhd.(2013)</p><p>EXTRA QUESTIONS</p><p>1. The weights of mangoes from a certain orchard follows a normal distribution. A quarter of the mangoes weigh less than 70 g and a third weigh more than 120 g. Find the mean weight and standard deviation of the mangoes produced by the orchard.</p><p>2. An orchard markets and sells fresh mangoes. Mangoes weighing less than 38 g are graded</p><p> small, those weighing more than 49 g are graded large and the rest are graded medium. The</p><p> weight of a mango from a particular tree is normally distributed with mean 42 g and</p><p> standard deviation 4 g. Find</p><p> a) the proportion of mangoes graded small</p><p> b) the proportion of mangoes graded medium</p><p>Page 3 of 10 c) the median weight of the mangoes graded large.</p><p>3. The mass of babies born in a hospital is normally distributed with a mean of 2.8 kg and a</p><p> standard deviation of 0.4 kg.</p><p> a) Find the probability that a baby born in the hospital has a mass between 2.6 kg and 3.1</p><p> kg.</p><p> b) Given that the mass of babies in the first quartile is m kg, determine the value of m.</p><p> c) If 10 babies who are born in the hospital is picked at random, calculate the probability</p><p> that at least one of the babies will have a mass exceeding 3.4 kg.</p><p>Page 4 of 10 4. The average number of milligrams (mg) of sodium in a certain brand of low-salt microwave</p><p> frozen dinners is 660 mg, and the standard deviation is 35 mg. Assume the variable is</p><p> normally distributed. </p><p> a) If a single dinner is selected, find the probability that the sodium content will be more</p><p> than 670 mg.</p><p> b) If a sample of 10 dinners is randomly selected, find the probability that the mean of the</p><p> sample will be larger than 670 mg.</p><p>Page 5 of 10 c) Why is the probability for part (a) greater than that for part (b)?</p><p>5. The average breaking strength of a certain brand of steel cable is 2000 pounds, with a</p><p> standard deviation of 100 pounds. A sample of 20 cables is selected and tested. Find the</p><p> sample mean that will cut off the upper 95% of all samples of size 20 taken from the</p><p> population. Assume the variable is normally distributed. </p><p>6. In a medical research, a new drug is being treated against a specific disease on humans. Clinical tests show that the probability a patient with the disease is cured by taking the new drug is 0.75. a) If 10 patients are treated with the new drug, find the probability that at most 6 of them will be cured. b) If 1000 patients are treated with the new drug, find i) the probability that 758 to 778 patients will be cured. ii) the value of n such that the probability that at least n patients will be cured is 0.8. </p><p>Page 6 of 10 7. The number of bacteria on a plate viewed under a microscope follows a Poisson</p><p> distribution with a mean of 60. </p><p> a) Find the probability that there are between 55 and 75 bacteria on a plate.</p><p> b) A plate is rejected if less than 38 bacteria are found. If 2000 such plates are viewed, how</p><p> many will be rejected?.</p><p>8. Tomatoes from a particular nursery are packed in boxes and sent to a market. Assume that the number of bad tomatoes in a box has a Poisson distribution with mean 0.44. a) Find the probability of there being two or more bad tomatoes in a box when it is opened. b) Using a suitable approximation, calculate the probability that in 80 randomly chosen boxes there will be fewer than 20 bad tomatoes in total.</p><p>9. Prepaid phone cards produced by a factory are packed in boxes. Each box contains 100 prepaid phone cards. It is known that 3% of the prepaid phone cards produced are defective. a) Show that the probability that a box chosen at random will contain at most 2 defective prepaid phone cards is approximately 0.42. b) If 15 boxes are chosen at random, find the probability that 6 boxes will contain at most 2 defective prepaid phone cards. c) Eighty boxes are chosen at random. Calculate the probability that between 30 and 50, inclusive, boxes will contain at most 2 defective prepaid phone cards.</p><p>Page 7 of 10 10. A batik painter knows from past experience that when painting batik cloths, paint spots</p><p>2 occur on the cloth at random and at a rate of 3 for every 900 cm . The batik painter wishes</p><p> to paint a piece of cloth 35 cm by 30 cm. </p><p> a) Find the probability that after painting the cloth, the cloth will contain</p><p> i) exactly 6 paint spots,</p><p> ii) not more than 4 paint spots.</p><p> b) Eight of these cloths are painted consecutively. Find the probability that 3 of them will</p><p> each contain exactly 4 paint spots.</p><p>Page 8 of 10 c) It is known that the painter will reject a paint job if there are more than 4 paint spots</p><p> found on the cloth. Using a suitable approximation, find the probability that out of 40</p><p> pieces of cloth, more than 10 pieces will be rejected.</p><p>Among the best of you [are they] who have the best character. The Prophet Muhammad (may Allah bless him and grant him peace)</p><p>Questions from textbook:</p><p>Sectio Question Answer Section Question Answer n 8.2 13 0.3962, 0.9771, 0.0228 8.5 5 0.1964, 0.1985 14 0.6826, 39.66 7 0.8485, 0.0985 8.3 3 0.7745, 60.6%, 0.0228 8 0.9798, 0.6063 4 0.82% 11 0.8444, 0.2457, 0.6406 5 0.6272, 9.17, 89 Review 6 A=75, B=74, 62 7 5, 40, 0.9759 8 0.9279, 0.1056 8.4 6 0.2135, 0.3085 10 0.2389, 20, 0.1112 12 0.9428, 0.9822, 0.0571 12 0.9387, 0.0052, 81.51</p><p>Page 9 of 10 13 0.5015, 27, 0.9049</p><p>Extra Questions:</p><p>Question Answer 1 100.18, 45.04 2 a) 0.1587 b) 0.8012 c) 50.2 3 a) 0.4649 b) 2.532 c) 0.4991 4 a) 0.3859 b) 0.1841 5 1963.10 6 a) 0.2241 b) i) 0.2724 ii) 739 7 a) 0.6883 b) 4 8 a) 0.0726 b)0.004 9 b) 0.2041 c) 0.8237 10 a) i) 0.0771 ii) 0.7254 b) 0.1324 c) 0.5675</p><p>Page 10 of 10</p>