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BusinessStatisticsChapter7(Fall2012).doc

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Department
Quantitative Methods
Course
QMS 102
Professor
Jason Chin- Tiong Chan
Semester
Fall

Description
QMS102-Business Statistics I Chapter7 Chapter 7 Continuous Random Variable - Normal Distributions Outcomes: • Reveal the characteristics of the Normal Distribution • Recognize the Standard Normal Distribution • Calculate the normal probabilities Characteristic of Normal Distribution • bell-shaped and has a single peak. • is symmetric about the mean. • is asymptotic, that is, the curve gets closer and closer to the x-axis but never actually touches it. • the total area under the curve is 1 • is denoted by N(μ,σ) The Standard Normal Distribution • with a mean of 0 and a standard deviation of 1, N(0, 1). • Z-value or Z-score – the units marked on the horizontal axis of the standard normal curve and are called z values or z scores. • the z values on the right side of the mean are positive • the z values on the left side of the mean are negative • the standard normal distribution table Fall2012 Page #1 QMS102-Business Statistics I Chapter7 Standardizing a Normal Distribution, Z • If the random variable X has a normal distribution with mean μ and standard deviation σ, that are different from 0 and 1 respectively. • Convert the given normal distribution to the standard normal distribution • Convert the random variable X to the standard normal variable Z, using the equation z = x −μ σ • Thus, P(X < x) = P(Z < z), where z is called the z-value. Example1 Let x be a normal random variable with its mean equal to 40 and standard deviation equal to 5. Determine the following probabilities: a. P (x ≥ 50) b. P (x < 47) c. P (42 ≤ x ≤ 49) Example2 Let x be a normal random variable with its mean equal to 12 and variance equal to 4. Determine the following probabilities: a. P (x ≥12 ) b. P (x <12.8) c. P (9.24 ≤ x ≤11.8) Fall2012 Page #2 QMS102-Business Statistics I Chapter7 Example3 a. What is the area under the normal curve between z = –2.34 and z = –1.45? b. What is the area under the normal curve between z = -9.0 and z = 0? c. Determine i. P(−1.45 ≤ z ≤ 2.06) ii.P z ≥ −1.47) Example4 The monthly credit card bills for households in Toronto are normally distributed, with mean of $1476.30 and the standard deviation of $385.89. a. Define the random variable X b. Determine the probability that a randomly selected bill is i. less than $1600 ii. more than $1200 iii. between $1100 and $1700 Example5 The recent average starting salary for new college graduates in marketing is $38,000. Assume salaries are normally distributed with a standard deviation $3500. a. Define the random variable X b. Determine the probability of a new graduate receiving a salary of i.more than $39,000 ii.less than $45,000 iii.. between $35,000 and $40,000 iv.between $$38,000 and $46,000 Fall2012 Page #3 QMS102-Business Statistics I Chapter7 Example6 Ryerson Trucking Company determined that on an annual basis the distance traveled per truck is normally distributed with a mean of 54.6 thousand miles and a standard deviation of 11.8 thousand miles. a. What percentage of trucks can be expected to travel either below 32 or
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