1. MBA 510 Lecture 4
Spring 2013
Dr. Tonya Balan
10/30/2019

2. Continuous Random Variables
If a random variable is continuous, its probability distribution is called a continuous probability distribution. A continuous probability distribution differs from a discrete probability distribution as follows:
The probability that a continuous random variable will assume a particular value is zero.
An equation or formula is used to describe a continuous probability distribution (PDF)
The PDF can be represented graphically
The area under the PDF is 1
The probability that a random variable assumes a value between a and b is equal to the area under the curve between a and b.
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3. Normal Distribution
The Normal Distribution is the most important of all continuous probability distributions and plays a key role in many of the statistical methods that will be discussed in this class.
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4. Normal Distribution
If all possible values of X follow an assumed normal curve, then X is said to be a normal random variable and the population is normally distributed.
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5. Normal Distribution
The normal distribution can be completely defined by two parameters – the mean (µ) and the standard deviation (σ). The parameters describe the population and can be estimated by the sample statistics 𝑥 and s.
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7. Standard Normal Distribution
A random variable, X, is said to have a standard normal distribution if it is normally distributed with mean 0 and standard deviation 1. In general, for any normal random variable, X, with mean μ and standard deviation σ, 𝑍= 𝑋 − 𝜇 𝜎 is a standard normal random variable. This process is called “standardizing” the random variable.
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8. Example (using the table)
Suppose that X is a standard normal random variable. Use the table distributed in class to find the following probabilities: P(X ≤ 1.23) P(X ≥ 0.54) P(-0.78 ≤ X ≤ 0.27)
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9. Calculating Probabilities for Any Normal Variable
Sketch the normal curve
Shade the region of interest and mark the delimiting x-values
Compute the z-scores for the delimiting x-values found in step 2
Use the standard normal table to obtain the probabilities
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10. Example
Intelligence quotients (IQs) measured on the Stanford Revision of the Binet-Simon Intelligence Scale are known to be normally distributed with a mean of 100 and a standard deviation of 16. What is the probability that a randomly selected individual will have an IQ of less than 115? What is the probability that an individual will have an IQ between 115 and 140?
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11. Example
As reported by Runner’s World magazine, the times of the finishers in the New York City 10k run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes. Let X be the time of a randomly selected finisher. Find
P(X>75)
P(X<50 or X>70)
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12. Find the Z-Score
Suppose that you are given a probability for a normally distributed random variable. In order to find the (z) value corresponding to that probability:
Sketch the normal curve associated with the variable
Shade the region of interest
Use the table to obtain the z-score
Convert the z-score to an x-value using the following formula:
x = μ + σz
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13. Find the Z-Score - Example
Consider the previous example involving IQs. Obtain the 90th percentile for IQs. (Recall that the 90th percentile is the IQ that is higher than those of 90% of all people).
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14. Empirical Rule
Every normal distribution conforms to the following rule:
68% of the values fall within one standard deviation of the mean
95% of the values fall within two standard deviations of the mean
99.7% of the values fall within three standard deviations of the mean
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