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2.2 Normal Distributions Normal Distribution and Normal Curve A Normal Distribution is described by a Normal density curve. Any particular Normal Distribution is completely specified by 2 numbers: its mean and standard deviation. The mean of a Normal Distribution is at the center of the symmetric Normal Curve. The standard deviation is the distance from the center to the change of curvature points on either side. We abbreviate the Normal Distribution with mean π and standard deviation π π(π, π) 68-95-99.7 Rule In the Normal Distribution with mean π and standard deviation π: ο· Approximately 68% of the observations fall within π of the mean π ο· Approximately 95% of the observations fall within 2π of the mean π ο· Approximately 99.7% of the observations fall within 3π of the mean π Check Your Understanding The distribution of heights of young women aged 18 to 24 is approximately π(64.5, 2.5) 1. Sketch a Normal density curve for the distribution of young womenβs heights. Label the points one, two, and three standard deviations from the mean. 2. What percent of young women have heights greater than 67 inches? Show you work. 3. What percent of young women have heights between 62 and 72 inches? Show your work. Standard Normal Distribution The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. If a variable x has a Normal distribution π(π, π) Then the standardized variable π§ = π₯βπ π has the standard Normal distribution. 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. Check Your Understanding Use Table A in the back of the book to find the proportion of observations from a standard Normal distribution that fall in each of the following regions. In each case, sketch a standard Normal curve and shade the area representing the region. 1. π§ < 1.39 2. π§ > β2.15 3. β.56 < π§ < 1.81 Use Table A to find the value π from the standard Normal distribution that satisfies each of the following conditions. Sketch a curve representing your findings. 4. The 20th percentile 5. 45% of observations are > π§ How to Solve Problems Involving Normal Distributions State: Express the problem in terms of the observed variable x. Plan: Draw a picture of the distribution and shade the area of interest under the curve. Do: Perform calculations ο· Standardize and ο· Use Table A (or calculator) Conclude: Write your conclusion in the context of the problem. Check Your Understanding 1. Cholesterol levels above 240 mg/dl may require medical attention. What percent of 14 yr old boys have more than 240 mg/dl of cholesterol, if the distribution of cholesterol levels for 14 yr old boys is π(170, 30)? 2. People with cholesterol levels between 200 and 240 mg/dl are at considerable risk for heart disease. What percent of 14 yr old boys have blood cholesterol between 200 and 240 mg/dl? 3. What distance would a ball have to travel to be at the 80th percentile of Tiger Woodsβs drive lengths, if driving lengths had a mean of 304 yds and standard deviation of 8 yrds? Assessing Normality To determine whether your data is normally distributed, make a Normal probability plot. This is just a graph of the value of your variable in the problem against the z-score. If the Normal probability plot is approximately linear, your data is approximately Normally distributed. If not, it is not.