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The Normal Distribution

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.”

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The Normal Distribution

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  1. The Normal Distribution BUSA 2100, Sections 3.3, 6.2

  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.

  3. 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.

  4. 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.

  5. 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?

  6. 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.

  7. 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.

  8. 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?

  9. Normal Curve Example, p. 2 • (b) What is P(190 <= X <= 240)?

  10. 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.

  11. 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:

  12. 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?

  13. Backwards Normal Curve Example, Page 2 • Solve z = (X - mu) / sigma for X.

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