ECON 214 - The Normal Distribution.pdf

Slide 18 importance of the normal distribution

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Slide 18
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Importance of the Normal Distribution Measurements in many random processes are known to have distributions similar to the normal distribution. Normal probabilities can be used to approximate other probability distributions such as the Binomial and the Poisson. Distributions of certain sample statistics such as the sample mean are approximately normally distributed when the sample size is relatively large, a result called the Central Limit Theorem. Slide 19
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Importance of the Normal Distribution If a variable is normally distributed, it is always true that 68.3% of observations will lie within one standard deviation of the mean, i.e. X = µ ± σ 95.4% of observations will lie within two standard deviations of the mean, i.e. X = µ ± 2 σ 99.7% of observations will lie within three standard deviations of the mean, i.e. X = µ ± 3 σ Slide 20
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The normal curve showing the relationship between σ and μ Slide 21 m- 3s m-2s m-1s m m+1s m+2s m+ 3s
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Calculating probabilities using the standard normal Normal curves vary in shape because of differences in mean and standard deviation (see slide 16) . To calculate probabilities we need the normal curve (distribution) based on the particular values of µ and σ . However we can express any normal random variable as a deviation from its mean and measure these deviations in units of its standard deviation (You will see an example soon) . Slide 22
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Calculating probabilities using the standard normal That is, subtract the mean ( µ ) from the value of the normal random variable (X) and divide the result by the standard deviation ( σ ). The resulting variable, denoted Z , is called a standard normal variable and its curve is called the standard normal curve. The distribution of any normal random variable will conform to the standard normal irrespective of the values for its mean and standard deviation. Slide 23
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Calculating probabilities using the standard normal If X is a normally distributed random variable, any value of X can be converted to the equivalent value, Z , for the standard normal distribution by the formula Z tells us the number of standard deviations the value of X is from the mean. The standard normal has a mean of zero and variance of 1, i.e., Z ~ N(0, 1). Slide 24 X Z
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Calculating probabilities using the standard normal Tables for normal probability values are based on one particular distribution: the standard normal ; from which probability values can be read irrespective of the parameters (i.e. mean and standard deviation) of the distribution. Example - following from the illustration on slide 17, c onsider the height of women. How many standard deviations is a height of 175cm above the mean of 166cm?
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