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Econ 3790: Business and Economics Statistics

Econ 3790: Business and Economics Statistics. Instructor: Yogesh Uppal Email: yuppal@ysu.edu. Normal. p ( x ). x. Chapter 6 Continuous Probability Distributions. Normal Probability Distribution. Normal Probability Distribution.

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Econ 3790: Business and Economics Statistics

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  1. Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal Email: yuppal@ysu.edu

  2. Normal p(x) x Chapter 6 Continuous Probability Distributions • Normal Probability Distribution

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

  4. Normal Probability Distribution • It has been used in a wide variety of applications: Heights of people Scientific measurements

  5. Normal Probability Distribution • It has been used in a wide variety of applications: Test scores • Amounts • of rainfall

  6. Normal f (x) x x1 x2 Normal Distributions • The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the curve between x1and x2.

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

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

  9. Normal Probability Distribution • Characteristics The entire family of normal probability distributions is defined by itsmeanm and its standard deviations . Standard Deviation s x Mean m

  10. Normal Probability Distribution • Characteristics The mean can be any numerical value: negative, zero, or positive. The following shows different normal distributions with different means. x -10 0 20

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

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

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

  14. Standard Normal Probability 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. s = 1 z 0

  15. 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.

  16. Example: Air Quality • The mean PMI is 41 and the standard deviation is 20.5. • Suppose I want to find out the probability that air quality is good or what is the probability that PMI is greater than 50.

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