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The Normal Distribution BUSA 2100, Sections 3.3, 6.2 Introduction to the Normal Distribution The normal distribution is the most widely used probability distribution. Reason: Most variables observed in nature and many variables in business are “normally distributed.” The normal distribution is a bell-shaped curve. Characteristics of the Normal Distribution The normal distribution has 3 major characteristics. (1) Most important characteristic -Large frequencies near the mean, and small frequencies at the extremes. (2) It is symmetric about the mean. (3) It is infinite in extent (in theory) -doesn’t touch the x-axis. Examples of the Normal Distribution Women’s heights, men’s weights, IQs, daily sales. (Explain) The size of the standard deviation affects the shape of the normal curve. For all normal curves: 68+% of values are within +-1 std.dev.; 95+% of values are within +-2 std. dev.; 99+% of values are within +-3 std. dev. z-Values Definition: A z-value represents the number of standard deviations that an item is from the mean. A normal distribution has mean (mu) = 150 & std. dev. (sigma) = 20. Find the z-values for X = 170, 150, 130, & 195. What is the z-value for X = 163? What is the formula for z? z-Values, Page 2 A negative z-value means that the item is to the left of the mean. Items with z-values beyond +-3 std. deviations are called outliers. z-values, together with a normal curve table, can be used to find probabilities. Procedure for Calculating Normal Curve Probabilities Step 1: Draw a sketch. Step 2: Calculate the z-value(s). Step 3: Look up the table value(s) in a normal curve table. Step 4: Calculate the final answer. Normal Curve Example Example 1: Suppose that daily sales for a product are normally distributed with mean 220 and standard deviation 36. What is the approx. range for sales? (a) What is the prob. that sales are less than 268? Normal Curve Example, p. 2 (b) What is P(190 <= X <= 240)? Normal Curve Example, p. 3 (d) What is the probability that sales are larger than 265? The normal curve table measures all probabilities (areas) from the lower end. Types of Normal Curve Problems All the problems we have just done are called regular normal curve problems: Now we will do a backwards normal curve problem: Backwards Normal Curve Ex. Ex. Mileage for a tire is normally distributed with mean 36,500 and std. dev. 5,000. A customer refund will be given for the 10% of tires that get the least mileage. What mileage (X-value) qualifies? Backwards Normal Curve Example, Page 2 Solve z = (X - mu) / sigma for X.