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
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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)
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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
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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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