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1 The Normal Distribution Prepared by E.G. Gascon

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2 Properties of Normal Distribution Its peak occurs directly above the mean The curve is symmetric about the vertical line through the mean. The curve never touches the x-axis The area under the curve is always = 1. (This agrees with the fact that the sum of the probabilities in any distribution is 1.) Image text page 487 Peak

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3 Variations in Normal Curves One standard deviation is smaller than normal One standard deviation is equal to the normal One standard deviation is larger than normal

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4 The Area Under the Standard Normal Curve The area of the shaded region under a normal curve form a point A to B is the probability that an observed data value will be between A and B Between -1 and +1 standard deviations there is 68% of the region, therefore the probability of an observed data value being within 1 standard deviation is 68%, etc. 1 standard deviation Image from text p 487 AB

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5 Problem solved using the Standard Normal Curve The area under a normal curve to the left of x (the data) is the same as the area under the standard normal curve to the left of the z-score for x. What does that mean? The z-score is the formula that converts the raw data (x) from a normal distribution into the lookup values of a STANDARD NORMAL CURVE. [See table in appendix of text or use Excel function =NORMSDIST(Z)] Example: sales force drives an average of 1200 miles, with a standard deviations of 150 miles miles is the mileage in question. First find the z-score

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6 What is the probability that a salesperson drives less than 1600 miles? Ans: It is the area to the left of the standard normal curve. Look up 2.67 in the Table of Normal Distributions. There is a 99.62% probability that the salesperson drives less than 1600 miles. 2.67

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7 Using Table of the Normal Distribution Z = 2.67 Look up 2.6 in the row, and.07 in the column. The intersection is the area to the left, or probability Table found in text page A-1 back of book

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8 Or Use Excel function Enter: Results:

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9 What is the probability that a salesperson drives more than 1600 miles? Ans: It is the area to the right of the standard normal curve. Since you know the are to the left of 2.67, the area to the right must be =.0038, or.38% probability that a salesperson drives more than 1600 miles. 2.67

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10 What is the probability that a salesperson drives between 1200 and 1600 miles? Ans: The area to the left of 2.67 is already known, it is Between = =.4962 Find the z value for 1200,, then look it up in the table. The probability that a salesperson drives between 1200 and 1600 miles is 49.62% It is the difference between driving less than 1600 and less than 1200.

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11 Questions / Comments / Suggestions Please post questions, comments, or suggestions in the main forum regarding this presentation.

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