#### Transcript Section 2.2 First Day Normal Curves, the

+ Chapter 2: Modeling Distributions of Data Section 2.2 Normal Distributions The Practice of Statistics, 4th edition - For AP* STARNES, YATES, MOORE + The Empirical Rule a.k.a. 68-95-99.7 Rule All normal distributions follow this rule: 68% of the observations are within one standard deviation of the mean 95% of the observations are within two standard deviations of the mean 99.7% of the observations are within three standard deviations of the mean + Yay, Math! IQ scores on the WISC-IV are normally distributed with a mean of 100 and a standard deviation of 15. up one σ and down one σ from 100 gives us the range from 85 to 115. 68% of people have an IQ between 85 and 115. Going 95% of people have an IQ between ____ and ____. 99.7% of people have an IQ between ____ and ____. + Try This heights of women aged 18 – 24 is approximately normally distributed with a mean μ = 64.5 inches and a standard deviation σ = 2.5 inches. The Between what two heights does the middle 95% fall? The tallest 2.5% of women are taller than what? What is the percentile for a woman who is 64.5 inches tall? a) Sketch the Normal density curve for this distribution. b) What percent of ITBS vocabulary scores are less than 3.74? c) What percent of the scores are between 5.29 and 9.94? Normal Distributions The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55). + Example, p. 113 All Normal distributions are the same if we measure in units of size σ from the mean µ as center. Definition: The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized variable z x - has the standard Normal distribution, N(0,1). Normal Distributions Standard Normal Distribution + The + Standard Normal Table Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table. Definition: The Standard Normal Table Table A is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z. Suppose we want to find the proportion of observations from the standard Normal distribution that are less than 0.81. We can use Table A: Z .00 .01 .02 0.7 .7580 .7611 .7642 0.8 .7881 .7910 .7939 0.9 .8159 .8186 .8212 P(z < 0.81) = .7910 Normal Distributions The + Example, p. 117 Finding Areas Under the Standard Normal Curve Normal Distributions Find the proportion of observations from the standard Normal distribution that are between -1.25 and 0.81. + Using Table A There are four types of questions I can ask with Table A. Less than: Find P(Z<0.42) Greater than: Find P(Z>0.42) Between: Find P(0.42 < Z < 1.3) Backwards: What Z-score marks the 15th percentile? What Z-score has 62% of observations to the right?