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Download 2.5 – The Normal Distribution Name Example 2: Finding the z
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2.5 – The Normal Distribution I. The Standard Normal Distribution Has a mean of and a standard deviation of Name . The standard normal distribution is symmetric, with total area under the curve equal to 1, or 100%. To find the percentage, P, that describes the area to the left of the corresponding z-‐score, you can use the z-‐table (page 824-‐825 in textbook) or use your calculator. Example 1: Finding the Percentage when you know the z-‐score A. Find the percentage, P, of values less than z=1.23, the shaded area in Display 2.77. B. Find the percentage greater than z=1.23. Example 2: Finding the z-‐score when you know the percentage. Find the z-‐score that falls at the 75th percentile of the standard normal distribution, that is, the z-‐score that divides the bottom 75% of the values from the rest. D28. For the standard normal distribution, a. What is the median? b. What is the lower quartile? c. What z-‐score falls at the 95th percentile? d. What is the IQR? II. Standard Units (z-‐score): Converting to standard units, or standardizing, is the two-‐step process of re-‐centering and rescaling that turns any normal distribution into the standard normal distribution. 1. 2. Z-‐Score: Example 3: Computing a z-‐score In a recent year, the distribution of SAT I math scores for the incoming class at the University of Georgia was roughly normal, with a mean of 610 and standard deviation of 69. What is the z-‐score for a University of Georgia student who got 560 on the math SAT ? Example 4: Finding the value when you know the z-‐score What was a University of Georgia student’s SAT I math score if his/her score was 1.6 standard deviations above the mean? Example 5: Using z-‐scores to make a comparison In the US, heart disease kills roughly one-‐and-‐a-‐quarter times as many people as cancer. If you look at the death rate per 100,000 residents by state, the distributions will be roughly normal. Compute and compare the z-‐scores for the heart disease and cancer data. III. Solving the unknown percentage or unknown value problem. Example 6: Percentage of males taller than 74 inches For groups of similar individuals, heights often are approximately normal in their distribution. For example, the heights of 18-‐24 year old males in the US are approximately normal, with mean of 70.1 inches and standard deviation of 2.7 inches. What percentage of these males are more than 74 inches tall? Example 7: 75th percentile of female heights The heights of females in the US who are ages 18-‐24 are approximately normally distributed, with a mean of 64.8 inches and standard deviation of 2.5 inches. What height separates the shortest 75% from the tallest 25%? IV. Central Intervals for Normal Distributions 68% 95% 99.7%
