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Lecture: Normal Distribution

Lecture: Normal Distribution. Professor Catherine Comiskey Understanding the Normal Distribution. Normal Distribution Overview. Random variables, experiments and probability distributions Continuous data and the Normal distribution Examples and exercises. Random Experiments: Terminology.

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Lecture: Normal Distribution

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  1. Lecture: Normal Distribution Professor Catherine Comiskey Understanding the Normal Distribution

  2. Normal Distribution Overview • Random variables, experiments and probability distributions • Continuous data and the Normal distribution • Examples and exercises

  3. Random Experiments: Terminology • Interested in the population, the data we have is usually a sample. • Need the sample to be representative of the population, we achieve this by choosing a random sample. • A random experiment is the process of selecting a random sample from the population • Probability theory is the area of mathematics that deals with random experiments.

  4. Continuous Data • See following slides based on and adapted from Williams, Sweeny and Anderson.

  5. Continuous Probability Distributions • A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. • It is not possible to talk about the probability of the random variable assuming a particular value. • Instead, we talk about the probability of the random variable assuming a value within a given interval.

  6. 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.

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

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

  9.  = mean  = standard deviation  = 3.14159 e = 2.71828 Normal Probability Distribution • Normal Probability Density Function where:

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

  11. 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

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

  13. Normal Probability Distribution • Characteristics The mean can be any numerical value: negative, zero, or positive. x -10 0 20

  14. 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

  15. 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

  16. of values of a normal random variable are within of its mean. 68.26% +/- 1 standard deviation of values of a normal random variable are within of its mean. 95.44% +/- 2 standard deviations of values of a normal random variable are within of its mean. 99.72% +/- 3 standard deviations Normal Probability Distribution • Characteristics

  17. 99.72% 95.44% 68.26% Normal Probability Distribution • Characteristics x m m + 3s m – 3s m – 1s m + 1s m – 2s m + 2s

  18. Standard Normal Probability Distribution A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability distribution.

  19. Standard Normal Probability Distribution The letter z is used to designate the standard normal random variable. s = 1 z 0

  20. Standard Normal Probability Distribution • Converting to the Standard Normal Distribution We can think of z as a measure of the number of standard deviations x is from .

  21. Class Exercise According to the Central Statistics Office (2006) the average industrial wage is €600 per week with a standard deviation of €75. What proportion of industrial workers earned between €525 and €675 per week? What is the probability an industrial worker earned between €450 and €750 per week?

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