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The Normal Distribution WEEKS FOUR and FIVE This worksheet relates to chapter six of the text book (Statistics for Managers 4th Edition). Since the normal distribution topic is split over two weeks there are normal distribution questions in both weeks 4 and 5. DISCUSSION QUESTIONS 1. Discuss the following terms ¹ Probability density function ¹ Normal curve ¹ Standard normal curve 2. List four properties of the normal curve and then draw it. What is the area under the normal curve? 30 CALCULATION QUESTION For these sorts of questions it is very important to know exactly what you are looking for. Always draw a diagram! It will really help. RACQ has completed a survey and think tyres have a mean life of 25000kms and a standard deviation of 1600 kms. Assume that the distribution of tyre life is normally distributed. What is the probability that a tyre will last: a) at least 22600 km b) at most 27100 km 31 MULTIPLE CHOICE PRACTICE 1. The effective range of the normal distribution is (a) (b) (c) (d) 2.57 standard deviations 3 standard deviations 6 standard deviations mean ± 6 standard deviations 2. If X is a normally distributed random variable with a mean of 22 and variance of 25, then the probability that x is greater than 34.3 is: (a) (b) (c) (d) Zero 0.0069 0.4931 0.9931 3. If x is a normally distributed random variable with a mean of 12 and a standard deviation of 3. The probability that x = 19.62 is: (a) (b) (c) (d) 0.00554 0.49446 0.99446 0 4. If X is normally distributed with a mean of 12 and a standard deviation of 3, then P(9 ≤ x≤ 11) is (a) (b) (c) (d) 0.2120 0.6293 0.4706 0.2234 Final, June 2005 32 5. In a university statistics course a number of years ago, the top 8% of students were given a grade of 7. If the mean score of students is 61.1 and the standard deviation is 9.7, and the scores are normally distributed, what score would you need to obtain in order to get a seven? (a) (b) (c) (d) 74.78 84.47 68.74 69.25 Final, June 2005 6. A psychological “introvert-extrovert” test produced scores which had a normal distribution with mean and standard deviation of 75 and 12 respectively. We wish to designate the highest 15% as extrovert, what would be the proper score to choose as the cut-off point? (Round answer to the nearest whole number) (a) (b) (c) (d) 87 76 88 63 Final, June 2004 7. John East Canneries only select the smallest 20% of sardines for their factories. If the length of sardines is normally distributed with a mean of 30cms and a variance of 25cm2, what is the maximum length sardine acceptable to John East? (a) (b) (c) (d) 34.2cm 51cm 9 cm 25.8 cm 8. Which of the following statements is false? If the normal distribution of wages of a group of workers has a mean of $12.00 per hour, then the proportion of workers who earn more than $11.60 is greater than… (draw a diagram to help you!) (a) (b) (c) (d) the proportion earning less than $11.60 per hour the proportion earning less than $12.60 per hour 50% the proportion earning more than the median wage 33 notes 34