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 A continuous probability distribution differs from a discrete probability distribution in several ways. A continuous random variable can take on an infinite number of values. That is, the probability for a specific variable is always zero. As a result, a continuous probability distribution cannot be expressed in tabular form. Instead, an equation or formula is used to describe a continuous probability distribution.

Probability Density Function The equation for describing a continuous probability distribution is called a probability density function and noted as “f(x).” Sometimes refer to as Density function pdf or PDF

Probability Density Function Example The shaded area in the graph represents the probability that the random variable X is less than or equal to a. Cumulative probability Always to the left The probability that the random variable X  is exactly equal to a would be zero.

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

The Normal Distribution ‘Bell Shaped’ Symmetrical Mean=Median=Mode Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   f(x) σ x μ Mean Median Mode

Position and Shape of the Normal Curve

Basis for the empirical rules 68.26% of the area under the curve falls within +/- 1 Stdev. 95.44% of the area under the curve falls within +/- 2 Stdev. 99.73% of the area under the curve falls within +/- 3 Stdev.

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) 0.5 0.5 μ x

Normal Probability Distribution The normal distribution refers to a family of continuous probability distributions described by the following equation. Where:  = mean  = standard deviation  = 3.14159 e = 2.71828

The Standard Normal Distribution Also known as the “z” distribution Mean is defined to be 0 Standard Deviation is 1 f(z) 1 z Values above the mean have positive z-values Values below the mean have negative z-values

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: 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 scale. Several sections of BA 301 by different instructors. Distributions of scores might be different due to differences in teaching methods and grading procedures…. By calculating Z value for each student, the distributions of the Z-value would be approximately the same in each section. That is, it is very easy to interpret the result using the Z-value since it is standardized.

Example 2 Statistically (based standard normal distribution), are they really or fundamentally different matters? SAT mean 1020 with standard deviation of 160. You scored 1260. What is your percentile? Average adult male height is 70 inches with standard deviation of 2 inches. You are 73 inches tall. What is your height percentile?

Z-scores for Two Problems Both are 1.5 Std Dev above the mean…. Download the Normal Dist. using Excel file from the class website

Four Excel Functions NORMDIST(x, mean, standard_dev, cumulative) Gives the probability that a number falls at or below a given value of a normal distribution NORMSDIST(z) Translates the number of standard deviations (z) into cumulative probabilities

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 5' 8", that is, 68 inches (172.72 cm)? NORMDIST(68, 65.5, 2.5, TRUE)  =  84.13% STANDARDIZE(68, 65.5, 2.5) = 1, NORMSDIST(1) = 84.13% Cumulative probability (to the left on the normal curve) Thus, 1- 84.13% = 15.87% or NORMSDIST(-1) = 15.87%

Four Excel Functions NORMINV(probability, mean, standard_dev) Calculates the x variable given a probability NORMSINV(probability) Given the probability that a variable is within a certain distance of the mean, it finds the z value

Four Excel Functions 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 right figure shows)? NORMINV(0.25, 65.5, 2.5)  = 63.81 inches 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

Formula Table We have X Z p We want X=μ+zσ Norminv(p,μ,σ) Normsinv(p) Normdist(x,μ,σ,1) Normsdist(z)

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 0.4772 Example: P(0 < z < 2.00) = 0.4772 z 2.00

The Standard Normal Table (continued) The Standard Normal Table gives the probability between the mean and a certain z value The z value ALWAYS refers to the area between some value (-z or +z) and the mean Since the distribution is symmetrical, the Standard Normal Table only displays probabilities for ½ of the full distribution

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

General Procedure for Finding Probabilities Determine m and s Define the event of interest e.g., P(x > x1) Convert to standard normal Use the table to find the probability

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: 8 8.6 x 0.12 Z P(8 < x < 8.6) = P(0 < z < 0.12)

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 = 0  = 1 x z 8 8.6 0.12 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 Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) The probability of obtaining a value less than 8.6 P = 0.5 Z 8.0 8.6

Finding Normal Probabilities (continued) Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 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 Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x > 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 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

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

Recall Tchebyshev from Chpt. 3 Empirical Rules What can we say about the distribution of values around the mean if the distribution is normal? f(x) μ ± 1σ covers about 68% of x’s σ σ Recall Tchebyshev from Chpt. 3 x μ-1σ μ μ+1σ 68.26%

μ ± 2σ covers about 95% of x’s μ ± 3σ covers about 99.7% of x’s The Empirical Rule (continued) μ ± 2σ covers about 95% of x’s μ ± 3σ covers about 99.7% of x’s 3σ 3σ 2σ 2σ μ x μ x 95.44% 99.72%

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

Normal Probabilities in PHStat We can use Excel and PHStat to quickly generate probabilities for any normal distribution We will find P(8 < x < 8.6) when x is normally distributed with mean 8 and standard deviation 5

Select desired options and enter values PHStat Dialogue Box Select desired options and enter values

PHStat Output

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?)

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:

The Exponential Distribution Used to measure the time that elapses between two occurrences of an event (the time between arrivals) Examples: Time between trucks arriving at a dock Time between transactions at an ATM Machine Time between phone calls to the main operator Recall l = mean for Poisson (see Chpt. 5)

The Exponential Distribution The probability that an arrival time is equal to or less than some specified time a is where 1/ is the mean time between events and e = 2.7183 NOTE: If the number of occurrences per time period is Poisson with mean , then the time between occurrences is exponential with mean time 1/  and the standard deviation also is 1/l

Exponential Distribution (continued) Shape of the exponential distribution f(x)  = 3.0 (mean = .333)  = 1.0 (mean = 1.0) = 0.5 (mean = 2.0) x

Example Example: Customers arrive at the claims counter at the rate of 15 per hour (Poisson distributed). What is the probability that the arrival time between consecutive customers is less than five minutes? Time between arrivals is exponentially distributed with mean time between arrivals of 4 minutes (15 per 60 minutes, on average) 1/ = 4.0, so  = .25 P(x < 5) = 1 - e-a = 1 – e-(.25)(5) = 0.7135 There is a 71.35% chance that the arrival time between consecutive customers is less than 5 minutes

Using PHStat

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

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