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6.4 Applications of the Normal Distribution Objective: Find probabilities for a normally distributed variable by transforming it into a standard normal variable. Requirement to use the Standard Normal Distribution Curve The variable must be normally or approximately normally distributed In Chapter 6, assume the distributions meet this requirement. Using the Standard Normal Distribution Curve to Solve Problems Method 1. Transform the normal distribution into a STANDARD Normal Distribution using the formula: This process transforms the variable into z values. 2. Use the z value and area under the curve to solve the problem Solve the following using the Standard Normal Distribution process Example 1 If students’ scores for a test have a mean of 100 and a standard deviation of 15, find the percentage of scores that will fall below 112. Example 2 Each month, an American household generates an average of 28 pounds of newspaper for garbage recycling. Assume the standard deviation is two pounds. If a household is selected at random, find the probability of its generating between 27 and 31 pounds per month. Example 3 Using the scenario described in #1 above, find the probability of a household generating more than 30.2 pounds per month. Example 4 The American Automobile Association reports that the average time it takes to respond to an emergency call is 25 minutes. Assume the variable is approximately normally distributed and the standard deviation is 4.5 minutes. If 80 calls are randomly selected, approximately how many will be responded to in less than 15 minutes? 1 NOTE: When doing percentage problems, change the percentage to a decimal before multiplying and round the answer to the nearest whole number. Most items like this can’t be divided into parts. 6.4 Application of the Normal Distribution (Day 2) Finding Data Values Given Specific Probabilities Example 5 In order to qualify for a police academy, candidates must score in the top 10% on a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find the lowest possible score to qualify. Assume test scores are normally distributed. You can also use an alternative formula when looking for the value of the variable X X=z∙σ+μ Example 5 For a medical study, a researcher wishes to select people in the middle 60% of the population based on blood pressure. If the mean systolic blood pressure is 120 and the standard deviation is 8, find the upper and lower readings that would qualify people to participate in the study. Determining Normality Method 1 Draw a histogram and determine if it looks normal Method 2 Use Pearson’s Index of Skewness, PI If -1 < PI < 1, the distribution is approximately normal PI 3( X median) s s – standard deviation Method 3 Check for outliers. If a value is more than 1.5(Q3 – Q1) below Q1 or 1.5(Q3– Q1) above Q3, then the distribution is skewed. 2 Example 6 A survey of 18 high-technology firms showed that the number of days’ inventory they had on hand. Determine if the data are approximately normally distributed. 5 81 29 88 34 91 44 97 45 98 63 113 68 118 74 151 74 158 Example 7 The data shown consist of the number of games played each year in the career of Baseball Hall of Famer Bill Mazeroski. Determine if the data are approximately normally distributed. 81 159 163 148 142 143 152 34 67 135 162 112 151 130 70 152 162 3