QTM 4 - Statistical Distributions - Tutorial Solutions

# QTM 4 - Statistical Distributions - Tutorial Solutions -...

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1 QTM 4 Statistical Distributions Tutorial Questions& Solutions Q1. Consider a standard normally distributed random variable, z ~ N(0,1). Calculate the following in percentage terms; a) Pr( z < 1) b) Pr( z > 3) c) Pr( z > -1) d) Pr( z > -2) e) Pr(-1 < z < 1) f) Pr( z < -3 or z > 3). Solutions Q2. Consider a standard normally distributed random variable, z ~ N(0,1). Determine the value of a such that a)Pr( z < a ) = 50% b)Pr( z < a ) = 75% c)Pr(- a < z ) = 95% d)Pr(- a < z < a ) = 95% Solutions a) 84.13% b) 0.135% c) 84.13% d) 97.72% e) 68.27% f) 0.27%

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2 Q3. Suppose a particular business has an expected turnover of £1,000,000 per month with a standard deviation of £100,000. Let the level of turnover be denoted t . If the turnover is normally distributed; a) What is the probability that the turnover this month will be greater than £1,200,000? i.e. What is Pr( t > 1,200,000)? b) What is the probability that the turnover will be less than 750,000? i.e. Pr( t < 750,000)? c) Calculate Pr(850,000 < t < 1,100,000) Solutions Q4 A business is trying to decide whether to invest in a new factory. Expected production is 200,000 units a week, with a (sample) standard deviation of 40,000 units. Let level of production be denoted p . If the turnover is normally distributed: a) Calculate the probability that production will be greater than 300,000 units next week. a) 0 b) 0.674 c) -1.645 d) 1.96 By symmetry Mean 1,000,000 Standard Deviation 100,000 a) Z-score = 2 Probability 2.3% b) Z-score = -2.5 Probability 0.6% c) Lower Z-score = -1.5 Upper Z-score = 1 Probability = 77.45%
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