Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal

Chapter 6 Continuous Probability Distributions Normal Probability Distribution x p(x)p(x)p(x)p(x) Normal

The normal probability distribution is the most important distribution for describing a continuous random variable. It is widely used in statistical inference.

Heights of people Heights Normal Probability Distribution n It has been used in a wide variety of applications: Scientific measurements measurementsScientific

Amounts of rainfall Amounts Normal Probability Distribution n It has been used in a wide variety of applications: Test scores scoresTest

Normal Distributions n The probability of the random variable assuming a value within some given interval from x 1 to x 2 is defined to be the area under the curve between x 1 and x 2. x f ( x ) Normal x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2

The distribution is symmetric; its skewness The distribution is symmetric; its skewness measure is zero. measure is zero. The distribution is symmetric; its skewness The distribution is symmetric; its skewness measure is zero. measure is zero. Normal Probability Distribution n Characteristics x

The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. Normal Probability Distribution n Characteristics x Mean = 

The entire family of normal probability The entire family of normal probability distributions is defined by its mean  and its distributions is defined by its mean  and its standard deviation . standard deviation . The entire family of normal probability The entire family of normal probability distributions is defined by its mean  and its distributions is defined by its mean  and its standard deviation . standard deviation . Normal Probability Distribution n Characteristics Standard Deviation  Mean  x

Normal Probability Distribution n Characteristics The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. The following shows different normal zero, or positive. The following shows different normal distributions with different means. The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. The following shows different normal zero, or positive. The following shows different normal distributions with different means. x

Normal Probability Distribution n Characteristics  = 15  = 25 The standard deviation determines the width of the curve: larger values result in wider, flatter curves. The standard deviation determines the width of the curve: larger values result in wider, flatter curves. x Same Mean

Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Normal Probability Distribution n Characteristics.5.5 x Mean 

n Z-scores can be calculated as follows: Standardizing the Normal Values or the z-scores z-scores We can think of z as a measure of the number ofWe can think of z as a measure of the number of standard deviations x is from .

 0 z A standard normal distribution is a normal distribution with mean of 0 and variance of 1. If x has a normal distribution with mean (μ) and Variance (σ), then z is said to have a standard normal distribution. A standard normal distribution is a normal distribution with mean of 0 and variance of 1. If x has a normal distribution with mean (μ) and Variance (σ), then z is said to have a standard normal distribution. Standard Normal Probability Distribution

Example: Air Quality I collected this data on the air quality of various cities as measured by particulate matter index (PMI). A PMI of less than 50 is said to represent good air quality. The data is available on the class website. Suppose the distribution of PMI is approximately normal.

The mean PMI is 41 and the standard deviation is Suppose I want to find out the probability that air quality is good or what is the probability that PMI is greater than 50. Example: Air Quality