Normal Distributions - Washington-Liberty

Section 2.2 Notes - Almost Done

Section 2.2: The Normal Distributions

Normal Distributions

A class of distributions whose density curves are symmetric, uni-modal, and bell-shaped. Normal distributions are VERY important in statistics. Which numerical summary would we use to describe the center and spread of a Normal distribution? Notation:

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Calculating using the Normal density curve

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The 68-95-99.7 Rule - In the Normal distribution with mean ad standard deviation :

? 68% of all the observations fall within one standard deviation () of the mean (in both directions)

? 95% of all the observations fall within two standard deviations (2) of the mean (in both directions)

? 99.7% of all the observations fall within three standard deviations (3) of the mean (in both directions)

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The distribution of heights of women aged 20 to 29 is approximately Normal with mean 64 inches and standard deviation 2.7 inches. Use the 68-95-99.7 rule to answer the following questions. (a) Between what heights do the middle 95% of young women fall?

(b) What percent of young women are taller than 61.3 inches?

You try: The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. Use the 68-95-99.7 rule to answer the following questions. (a) Between what values do the lengths of the middle 68% of all pregnancies fall?

(b) How short are the shortest 2.5% of all pregnancies?

(c) What percent of pregnancies are longer than 314 days?

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The standard Normal distribution

? InXinitely many Normal distributions One for every possible combination of means and standard deviations

? Standard Normal distribution - N(0, 1)

? We can standardize any value of a variable, x. This standardized value is called the z-score, or z. If we actually want to do calculations using this standardized score we need to know the distribution of the original variable. If the original variable is Normal then the z-score comes from a standard Normal distribution.

? A z-score tells us how many standard deviations the original observation falls away from its mean AND in which direction.

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