Yandell – Econ 216 Chap 6-1 Chapter 6 The Normal Distribution and Other Continuous Distributions

Yandell – Econ 216 Chap 6-2 Chapter Goals After completing this chapter, you should be able to: Describe the characteristics of the normal distribution Translate normal distribution problems into standard normal distribution problems Find probabilities using a normal distribution table and apply the normal distribution to business situations Use PHStat to obtain normal probabilities Recognize when to apply the uniform and exponential distributions

Yandell – Econ 216 Chap 6-3 Probability Distributions Continuous Probability Distributions Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Normal Uniform Exponential Ch. 5Ch. 6

Yandell – Econ 216 Chap 6-4 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.

Yandell – Econ 216 Chap 6-5 The Normal Distribution Continuous Probability Distributions Normal Uniform Exponential

Yandell – Econ 216 Chap 6-6 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   Mean = Median = Mode X f(X) μ σ

Yandell – Econ 216 Chap 6-7 By varying the parameters μ and σ, we obtain different normal distributions Many Normal Distributions listen

Yandell – Econ 216 Chap 6-8 The Normal Distribution Shape X f(X) μ σ Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread.

Yandell – Econ 216 Chap 6-9 Finding Normal Probabilities Probability is the area under the curve! ab X f(X) PaXb( )  Probability is measured by the area under the curve

Yandell – Econ 216 Chap 6-10 f(X) X μ Probability as Area Under the Curve 0.5 The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below

Yandell – Econ 216 Chap 6-11 Empirical Rules μ ± 1 σ  encloses about 68% of X’s  f(X) X μ μ  σμ  σ What can we say about the distribution of values around the mean? There are some general rules: σσ 68.26%

Yandell – Econ 216 Chap 6-12 The Empirical Rule μ ± 2σ covers about 95% of X’s μ ± 3σ covers about 99.7% of X’s xμ 2σ2σ2σ2σ xμ 3σ3σ3σ3σ 95.44%99.72% (continued)

Yandell – Econ 216 Chap 6-13 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

Yandell – Econ 216 Chap 6-14 Inference Put another way, when a value of X is observed that is far from a hypothesized value for the mean, because this observation is so unlikely if the hypothesis is true, it casts strong doubt on the claim about the mean. (More in Chapters 7 and 8)

Yandell – Econ 216 Chap 6-15 The Standard Normal Distribution Also known as the “Z” distribution Mean is defined to be 0 Standard Deviation is 1 Z f(Z) 0 1 Values above the mean have positive Z-values, values below the mean have negative Z-values

Yandell – Econ 216 Chap 6-16 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

Yandell – Econ 216 Chap 6-17 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:

Yandell – Econ 216 Chap 6-18 Example If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100.

Yandell – Econ 216 Chap 6-19 Comparing X and Z units Z X 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) μ = 100 σ = 50

Yandell – Econ 216 Chap 6-20 The Standard Normal Table The Standard Normal table in the textbook (Appendix table E.2) gives the probability less than a desired value for Z (i.e., from negative infinity to Z) Z Example: P(Z < 2.00) =.9772

Yandell – Econ 216 Chap 6-21 The Standard Normal Table The value within the table gives the probability from Z =   up to the desired Z value P(Z < 2.00) =.9772 The row shows the value of Z to the first decimal point The column gives the value of Z to the second decimal point (continued)

Yandell – Econ 216 Chap 6-22 General Procedure for Finding Probabilities Draw the normal curve for the problem in terms of X Translate X-values to Z-values Use the Standard Normal Table To find P(a < X < b) when X is distributed normally:

Yandell – Econ 216 Chap 6-23 Finding Normal Probabilities Suppose X is normal with mean 8.0 and standard deviation 5.0 Find P(X < 8.6) X

Yandell – Econ 216 Chap 6-24 Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6) P(Z < 0.12) Z X P(X < 8.6) μ = 8 σ = 5 μ = 0 σ = 1 (continued) Finding Normal Probabilities

Yandell – Econ 216 Chap 6-25 Z 0.12 Z Solution: Finding P(Z < 0.12) Standard Normal Probability Table (Portion) 0.00 = P(Z < 0.12) P(X < 8.6)

Yandell – Econ 216 Chap 6-26 Upper Tail Probabilities Suppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(X > 8.6) X

Yandell – Econ 216 Chap 6-27 Now Find P(X > 8.6)… (continued) Z Z =.4522 P(X > 8.6) = P(Z > 0.12) = P(Z < 0.12) = =.4522 Upper Tail Probabilities

Yandell – Econ 216 Chap 6-28 Probability Between Two Values Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(8 < X < 8.6) P(8 < X < 8.6) = P(0 < Z < 0.12) Z X8.6 8 Calculate Z-values:

Yandell – Econ 216 Chap Z 0.12 Z Solution: Finding P(0 < Z < 0.12) Standard Normal Probability Table (Portion) 0.00 = P(0 < Z < 0.12) P(8 < X < 8.6) = P(Z < 0.12) – P(Z < 0) = =

Yandell – Econ 216 Chap 6-30 Lower Tail Probabilities Suppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < X < 8) X

Yandell – Econ 216 Chap 6-31 Lower Tail Probabilities Now Find P(7.4 < X < 8)… X P(7.4 < X < 8) = P(-0.12 < Z < 0) = P(Z < 0) – P(Z < -0.12) = =.0478 (continued) Z The Normal distribution is symmetric, so this probability is the same as P(0 < Z < 0.12)

Yandell – Econ 216 Chap 6-32 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

Yandell – Econ 216 Chap 6-33 PHStat Dialogue Box Select desired options and enter values

Yandell – Econ 216 Chap 6-34 PHStat Output

Yandell – Econ 216 Chap 6-35 PHStat Demo Click below to view a PhStat demo to see how to get P(5 < X < 6.2), if μ = 5 and σ = 10: Click here to view normal distribution demo

Yandell – Econ 216 Chap 6-36 Upper Tail Practice Problem What value c from a standard normal distribution cuts off 5% in the upper tail? i.e., what is c so that P(Z > c) =.05 ? Z C 0.05

Yandell – Econ 216 Chap Upper Tail Practice Problem Z C = (continued) Z Standard Normal Probability Table (Portion).9500 falls between Z = 1.64 and Z = 1.65, so approximate by choosing Z = 1.645

Yandell – Econ 216 Chap 6-38 Practice Problem Demo Click here to open the excel sheet to see the details (Z_table_upper5pct.xls) Click here to view the demonstration

Yandell – Econ 216 Chap 6-39 The Uniform Distribution Continuous Probability Distributions Normal Uniform Exponential

Yandell – Econ 216 Chap 6-40 The Uniform Distribution The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable

Yandell – Econ 216 Chap 6-41 The Continuous Uniform Distribution: where f(X) = value of the density function at any X value a = lower limit of the interval b = upper limit of the interval The Uniform Distribution (continued) f(X) =

Yandell – Econ 216 Chap 6-42 Uniform Distribution Example: Uniform Probability Distribution Over the range 2 ≤ X ≤ 6: f(X) = =.25 for 2 ≤ X ≤ X f(X)

Yandell – Econ 216 Chap 6-43 The Exponential Distribution Continuous Probability Distributions Normal Uniform Exponential

Yandell – Econ 216 Chap 6-44 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 an unloading dock Time between transactions at an ATM Machine Time between phone calls to the main operator

Yandell – Econ 216 Chap 6-45 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 Note that if the number of occurrences per time period is Poisson with mean, then the time between occurrences is exponential with mean time 1/

Yandell – Econ 216 Chap 6-46 Exponential Distribution Shape of the exponential distribution (continued) f(X) X = 1.0 (mean = 1.0) = 0.5 (mean = 2.0) = 3.0 (mean =.333)

Yandell – Econ 216 Chap 6-47 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) =.7135

Yandell – Econ 216 Chap 6-48 Chapter Summary Introduced the Normal probability distribution Found probabilities using formulas and tables Discussed the Standard Normal Distribution and applied it to decision problems Reviewed other continuous distributions uniform, exponential