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Introduction to Statistics Chapter 6 Continuous Probability Distributions Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-1 Chapter Goals After completing this chapter, you should be able to: Understand the continuous random variables and there probability distribution Recognize when to apply the uniform Find probabilities using a normal distribution table and apply the normal distribution Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-2 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-3 The Uniform Distribution The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-4 The Uniform Distribution (continued) The Continuous Uniform Distribution: f(x) = 1 ba if a x b 0 otherwise where f(x) = value of the density function at any x value a = lower limit of the interval b = upper limit of the interval Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-5 Uniform Distribution Example: Uniform Probability Distribution Over the range 2 ≤ x ≤ 6: 1 f(x) = 6 - 2 = .25 for 2 ≤ x ≤ 6 f(x) .25 2 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. 6 x Chap 5-6 The Normal Distribution ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: + to Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. f(x) σ x μ Mean = Median = Mode Chap 5-7 Many Normal Distributions By varying the parameters μ and σ, we obtain different normal distributions Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-8 The Normal Distribution Shape f(x) Changing μ shifts the distribution left or right. σ μ Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Changing σ increases or decreases the spread. x Chap 5-9 Finding Normal Probabilities Probability is the Probability is measured area under the curve! under the curve f(x) by the area P (a x b) a Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. b x Chap 5-10 Probability as Area Under the Curve The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below f(x) P( x μ) 0.5 0.5 P(μ x ) 0.5 0.5 μ x P( x ) 1.0 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-11 Empirical Rules What can we say about the distribution of values around the mean? There are some general rules: f(x) μ ± 1σ encloses about 68% of x’s σ μ1σ σ μ μ+1σ x 68.26% Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-12 The Empirical Rule (continued) μ ± 2σ covers about 95% of x’s μ ± 3σ covers about 99.7% of x’s 2σ 3σ 2σ μ x 95.44% Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. 3σ μ x 99.72% Chap 5-13 Importance of the Rule If a value is about 2 or more standard deviations away from the mean in a normal distribution, then it is far from the mean The chance that a value that far or farther away from the mean is highly unlikely, given that particular mean and standard deviation Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-14 The Standard Normal Distribution Also known as the “z” distribution Mean is defined to be 0 Standard Deviation is 1 f(z) 1 0 z Values above the mean have positive z-values, values below the mean have negative z-values Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-15 The Standard Normal Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (z) Need to transform x units into z units Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-16 Translation to the Standard Normal Distribution Translate from x to the standard normal (the “z” distribution) by subtracting the mean of x and dividing by its standard deviation: x μ z σ Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-17 Example If x is distributed normally with mean of 100 and standard deviation of 50, the z value for x = 250 is x μ 250 100 z 3.0 σ 50 This says that x = 250 is three standard deviations (3 increments of 50 units) above the mean of 100. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-18 Comparing x and z units μ = 100 σ = 50 100 0 250 3.0 x z Note that the distribution is the same, only the scale has changed. We can express the problem in original units (x) or in standardized units (z) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-19 The Standard Normal Table The Standard Normal table in the textbook (Appendix D) gives the probability from the mean (zero) up to a desired value for z .4772 Example: P(0 < z < 2.00) = .4772 0 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. 2.00 z Chap 5-20 The Standard Normal Table (continued) The column gives the value of z to the second decimal point z The row shows the value of z to the first decimal point 0.00 0.01 0.02 … 0.1 0.2 . . . 2.0 .4772 P(0 < z < 2.00)2.0 = .4772 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. The value within the table gives the probability from z = 0 up to the desired z value Chap 5-21 General Procedure for Finding Probabilities To find P(a < x < b) when x is distributed normally: Draw the normal curve for the problem in terms of x Translate x-values to z-values Use the Standard Normal Table Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-22 Z Table example Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) Calculate z-values: x μ 8 8 z 0 σ 5 x μ 8.6 8 z 0.12 σ 5 8 8.6 x 0 0.12 Z P(8 < x < 8.6) = P(0 < z < 0.12) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-23 Z Table example (continued) Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) =8 =5 8 8.6 =0 =1 x P(8 < x < 8.6) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. 0 0.12 z P(0 < z < 0.12) Chap 5-24 Solution: Finding P(0 < z < 0.12) Standard Normal Probability Table (Portion) z .00 .01 P(8 < x < 8.6) = P(0 < z < 0.12) .02 .0478 0.0 .0000 .0040 .0080 0.1 .0398 .0438 .0478 0.2 .0793 .0832 .0871 Z 0.3 .1179 .1217 .1255 0.00 0.12 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-25 Finding Normal Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) Z 8.0 8.6 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-26 Finding Normal Probabilities (continued) Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) P(x < 8.6) .0478 .5000 = P(z < 0.12) = P(z < 0) + P(0 < z < 0.12) = .5 + .0478 = .5478 Z 0.00 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. 0.12 Chap 5-27 Upper Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x > 8.6) Z 8.0 8.6 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-28 Upper Tail Probabilities (continued) Now Find P(x > 8.6)… P(x > 8.6) = P(z > 0.12) = P(z > 0) - P(0 < z < 0.12) = .5 - .0478 = .4522 .0478 .5000 .50 - .0478 = .4522 Z 0 0.12 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Z 0 0.12 Chap 5-29 Lower Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < x < 8) Z 8.0 7.4 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-30 Lower Tail Probabilities (continued) Now Find P(7.4 < x < 8)… The Normal distribution is symmetric, so we use the same table even if z-values are negative: .0478 P(7.4 < x < 8) = P(-0.12 < z < 0) Z = .0478 8.0 7.4 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-31 Chapter Summary Reviewed key continuous distributions uniform and normal, Found probabilities using formulas and tables Recognized when to apply different distributions Applied distributions to decision problems Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-32