Chapter 6 Introduction to Continuous Probability Distributions Business Statistics: A Decision-Making Approach 8th Edition Chapter 6 Introduction to Continuous Probability Distributions

Chapter Goals After completing this chapter, you should be able to: Convert values from any normal distribution to a standardized z-score Find probabilities using a normal distribution table Apply the normal distribution to business problems Recognize when to apply the uniform and exponential distributions (not covered)

Continuous Probability Distributions Unlike a discrete probability distribution, A continuous random variable can take on an infinite number of values (the exact probability for a specific variable is always zero). That is, it always calculate cumulative probability. Finding an “exact” probability is impossible. As a result, a continuous probability distribution cannot be expressed in tabular form.

Types of Continuous Distributions Three types Normal Uniform (so easy…by yourself) Exponential (not cover)

The Normal Distribution ‘Bell Shaped’ Symmetrical Mean=Median Location is determined by the mean, μ Spread is determined by the standard deviation, σ f(x) σ x μ Mean Median

Probability of Normal Distribution Probability for 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) 0.5 0.5 μ x

Finding Normal Probabilities Probability is measured by the area under the curve because a continuous random variable can take on an infinite number of values. That is, it always calculates cumulative probability f(x) P ( a  x  b ) a b x Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Standard Normal (z) Distribution Way to find probabilities for a normal distribution Also known as the “z” distribution Mean is defined to be 0 and Standard Deviation is 1 f(z) 1 z Values above the mean have positive z-values Values below the mean have negative z-values

Formula Translate from x to the standard normal (the “z” distribution) by subtracting the mean of x and dividing by its standard deviation: z is the number of standard deviations units that x is away from the population mean

Example 1 If x is distributed normally with mean of 100 and standard deviation of 50, the z value for x = 250 is This says that x = 250 is three standard deviations (3 increments of 50 units) above the mean of 100.

Comparing x and z units 100 250 x 3.0 z μ = 100 σ = 50 3.0 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)

Why standardizing? To measure variable with different means and/or standard deviations on a single standardized scale. Example: Statistically, are below examples really or fundamentally different matters? SAT mean 1020 with standard deviation of 160. If your score is 1260, how good is your score statistically? Average adult male height is 70 inches with standard deviation of 2 inches. If your height is 73 inches, how tall are you statistically?

Z-scores for Two Problems Both are 1.5 Std Dev above the mean….

“STANDARDIZE” Excel Function STANDARDIZE(x, mean, standard_dev) Find the z value for a specified value, mean, and standard deviation. STANDARDIZE(1260, 1020, 160)  =  1.5 Solve previous problems using Excel: Download the Normal Dist. using Excel file from the class website and then select the first tab “Example 1”….

The Standard Normal Table Example: Probability when z value is 2 P(0 < z < 2.00) = 0.4772 0.4772 z 2.00

The Standard Normal Table (continued) The column gives the value of z to the second decimal point The row shows the value of z to the first decimal point The value within the table gives the probability from z = 0 up to the desired z value . 2.0 .4772 P(0 < z < 2.00) = 0.4772 2.0

z Table Example Find probability when x = 8.6 with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) Calculate z-values: 8 8.6 x 0.12 Z P(8 < x < 8.6) = P(0 < z < 0.12)

Solution: Finding P(0 < z < 0.12) Standard Normal Probability Table (Portion) P(8 < x < 8.6) = P(0 < z < 0.12) .02 z .00 .01 0.0478 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

Finding Normal Probabilities Now Find P(x < 8.6): less than 8.6 The probability of obtaining a value less than 8.6 P = 0.5 Z 8.0 8.6

Finding Normal Probabilities (continued) Now Find P(x < 8.6) less than 8.6 0.0478 0.5000 P(x < 8.6) = P(z < 0.12) = P(z < 0) + P(0 < z < 0.12) = 0.5000 + 0.0478 = 0.5478 Z 0.00 0.12

Upper Tail Probabilities: important Excel always calculate area to the left of the value (x or z) Now Find P(x > 8.6): greater than 8.6 Z 8.0 8.6

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) = 0.5000 - 0.0478 = 0.4522 0.0478 0.5000 0.4522 Z Z 0.12 0.12

Lower Tail Probabilities Now Find P(7.4 < x < 8) Between 7.4 and 8 Z 8.0 7.4

Lower Tail Probabilities (continued) Now Find P(7.4 < x < 8)…the probability between 7.4 and the mean of 8 The Normal distribution is symmetric, so we use the same table even if z-values are negative: P(7.4 < x < 8) = P(-0.12 < z < 0) = 0.0478 0.0478 Z 8.0 7.4

Different Situations Require Different Approaches (not just finding probability) We have X Z p We want X=μ+zσ Norminv(p,μ,σ) Normsinv(p) Normdist(x,μ,σ,1) Normsdist(z)

Four Excel Functions NORMDIST(x, mean, standard_dev, cumulative) Find the probability that a number falls at or below a given value of a normal distribution (draw a graph for the accuracy) x -- The value you want to test. mean -- The average value of the distribution.  standard_dev -- The standard deviation of the distribution. cumulative -- If FALSE or zero, returns the probability that x will occur; if TRUE or non-zero, returns the probability that the value will be less than or equal to x.

Four Excel Functions The distribution of heights of American women aged 18 to 24 is approximately normally distributed with a mean of 65.5 inches (166.37 cm) and a standard deviation of 2.5 inches (6.35 cm). What percentage of these women is taller than (or smaller than) 68 inches (172.72 cm)? NORMDIST(68, 65.5, 2.5, TRUE)  =  84.13% (smaller than): cumulative probability (to the left on the normal curve) Thus, 1- 84.13% = 15.87% (taller than)

Four Excel Functions NORMSDIST(z) Translate the number of standard deviations (z) into cumulative probabilities (draw a graph for the accuracy because Excel always calculate area to the left of the value (x or z)) z -- The value for which you want the distribution. Find less than 1: NORMSDIST(1) = 0.8413

Four Excel Functions NORMINV(probability, mean, standard_dev) Calculate the x variable given a probability How tall would a woman need to be if she wanted to be among the tallest 75% of American women (find 25% shorter instead as the figure shows)? NORMINV(0.25, 65.5, 2.5)  = 63.81 inches

Four Excel Functions NORMSINV(probability) Given the probability that a variable is within a certain distance of the mean, it finds the z value Find the z values that mark the boundary that is 25% less than the mean and 25% (75%) more than the mean. NORMSINV(0.25) = - 0.674 NORMSINV(0.75) = 0.674

The Uniform Distribution The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable Referred to as the distribution of “little information” Probability is the same for ANY interval of the same width Useful when you have limited information about how the data “behaves” (e.g., is it skewed left?)

Uniform Distribution Examples Review examples first Example 6-3 on page 250 Example 6-4 on page 251

The Uniform Distribution (continued) The Continuous Uniform Distribution: f(x) = where f(x) = value of the density function at any x value a = lower limit of the interval of interest b = upper limit of the interval of interest

The Mean and Standard Deviation for the Uniform Distribution The mean (expected value) is: The standard deviation is where a = lower limit of the interval from a to b b = upper limit of the interval from a to b

Steps for Using the Uniform Distribution Define the density function Define the event of interest Calculate the required probability f(x) x

Uniform Distribution Example: Uniform Probability Distribution Over the range 2 ≤ x ≤ 6: 1 f(x) = = .25 for 2 ≤ x ≤ 6 6 - 2 f(x) .25 x 2 6

Uniform Distribution Example: Uniform Probability Distribution Over the range 2 ≤ x ≤ 6:

Chapter Summary Reviewed key continuous distributions normal uniform Found probabilities using formulas and tables Recognized when to apply different distributions Applied distributions to decision problems