Normal Distributions - Washington-Liberty

[Pages:17]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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YOU TRY: The heights of women aged 20 to 29 are approximately Normal with mean 64 inches and standard deviation 2.7 inches: N(64, 2.7). Men the same age have mean height 69.3 inches with standard deviation 2.8 inches and follow an approximately Normal distribution: N(69.3, 2.8). What are the z-scores for a woman 6 feet tall and a man 6 feet tall? Say in simple language what information the z-scores give the actual heights do not.

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Using the Normal curve to determine proportions/percentiles (Beyond the 68-95-99.7 rule)

? The area under any Normal curve (or density curve for that matter) is equal to 1.

? If we want to know the proportion of observations that lie within a certain range of observation values we look for the area of the density curve between those two values (for ANY density curve not just Normal)

? We have a table that gives us these values for ONLY the standard Normal distribution.

Use table A to +ind the proportion of observations from a standard Normal distribution that satis+ies each of the following statements. In each case, sketch a standard Normal curve and shade the area under the curve that is the answer to the question.

(a) z < 2.66

(b) z > - 1.45

(c) -0.58 < z < 1.93

Since we can standardize ANY Normal distribution we can use this table for ANY Normal distribution.

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FOR EXAMPLE: Suppose that the heights of young women have a Normal distribution, N(64, 2.7). What proportion or percentage of all young women are less than 70 inches tall?

Using the same distribution from the last example, what proportion of women are greater than 60 inches tall?

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