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

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. 10/30/2019

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. 10/30/2019

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. 10/30/2019

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. 10/30/2019

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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. 10/30/2019

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) 10/30/2019

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 10/30/2019

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? 10/30/2019

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) 10/30/2019

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 10/30/2019

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). 10/30/2019

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 10/30/2019