The Normal Distribution Chapter 2

Continuous Random Variable A continuous random variable: Represented by a function/graph. Area under the curve represents the proportions of the observations Total area is exactly 1. How do we locate the median for a continuous random variable? the mean? The median is the value that divides the graph into equal area while the mean is the “balance” point.

Continuous Random Variable 1 .4(1) =0.4 0.4 0.5 What percent of the observations lie below 0.4? 40%

Continuous Random Variable 2 1.4(.5) =0.7 0.6 What proportion of the observations lie above 0.6? Notice, to find proportion for observation above, we can use the complement rule.

Continuous Random Variable 3 Where is the mean and median? How will the curve change as s changes?

Normal Distributions Symmetric, single-peaked, and mound-shaped distributions are called normal distributions Normal curves: Mound shaped Mean = Median The mean m and standard deviation s completely determine the shape

The Normal Curve Will finding proportions work differently than previous random variable examples? Empirical Rule Discovery

68% of observations fall within 1s of m

95% of observations fall within 2s of m

99.7% of observations fall within 3s of m

68-95-99.7 Rule Applet

68-95-99.7 Rule 34% 34% .15% 2.35% 13.5% 13.5% 2.35% .15% Applet

Percentiles? .15th 2.5th 16th 50th 84th 97.5th 99.85th 34% 34% 2.35% 13.5% 13.5% 2.35%

What’s Normal in Statistics? Normal distributions are good descriptions for real data allowing measures of relative position to be easily calculated (i.e. percentiles) Much of statistical inference (in this course) procedures area based on normal distributions FYI: many distributions aren’t normal

Distribution of dates is approximately normal with mean 1243 and standard deviation of 36 years. 1135 1171 1207 1243 1279 1315 1351

Assume the heights of college women are normally distributed with a mean of 65 inches and standard deviation of 2.5 inches. 57.5 60 62.5 65 67.5 70 72.5

What percentage of women are taller than 65 in.? 50% 57.5 60 62.5 65 67.5 70 72.5

What percentage of women are shorter than 65 in.? 50% 57.5 60 62.5 65 67.5 70 72.5

What percentage of women are between 62.5 in. and 67.5 in.? 68% 57.5 60 62.5 65 67.5 70 72.5

What percentage of women are between 60 in. and 70 in.? 95% 57.5 60 62.5 65 67.5 70 72.5

What percentage of women are between 60 and 67.5 in? 68% 13.5% 81.5% 57.5 60 62.5 65 67.5 70 72.5

What percentage of women are shorter than 70 in.? 50% 34% 13.5% 97.5% 57.5 60 62.5 65 67.5 70 72.5