Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-1 Chapter 6 The Normal Distribution and Other Continuous Distributions Statistics for Managers Using Microsoft ® Excel 4 th Edition

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-2 Chapter Goals After completing this chapter, you should be able to:  Describe the characteristics of the normal, uniform and exponential distributions  Recognize when to apply the normal, uniform and exponential distributions  Find normal, uniform and exponential probabilities  Evaluate the normality assumption

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-3 Chapter Goals After completing this chapter, you should be able to:  Understand the concept of a sampling distribution  Determine the mean and standard deviation for the sampling distribution of the mean, X  Determine the mean and standard deviation for the sampling distribution of the proportion, p s  Describe the Central Limit Theorem and its importance  Apply sampling distributions for both X and p s _ _ (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-4 Probability Distributions Continuous Probability Distributions Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Normal and Sampling Uniform Exponential Ch. 5Ch. 6

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-5 Continuous Probability Distributions  A continuous random variable is a variable that can assume any value on a continuum (can assume an infinite 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.

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-6 The Normal Distribution Probability Distributions Normal Uniform Exponential Continuous Probability Distributions

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-7 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

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-8 The Normal Probability Density Function  The formula for the normal probability density function is Wheree = the mathematical constant approximated by π = the mathematical constant approximated by μ = the population mean σ = the population standard deviation X = any value of the continuous variable

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-9 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

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-10 Finding Normal Probabilities Probability is the area under the curve! ab X f(X) PaXb( ) ≤ Probability is measured by the area under the curve ≤ PaXb( ) << = (Note that the probability of any individual value is zero)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-11 Finding Normal Probabilities  Draw the normal curve for the problem in terms of X  Suppose X is normal with mean 8.0 and standard deviation 5.0  Find P(X < 8.6)  Use the Normal function in Excel or PhStat X

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-12 Upper Tail Probabilities  Suppose X is normal with mean 8.0 and standard deviation 5.0.  Now Find P(X > 8.6)  P(X > 8.6)=1- P(X < 8.6) X

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-13 Probability Between Two Values  Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(8 < X < 8.6)  = P(X < 8.6)- P(X < 8) P(8 < X < 8.6) X 8.6 8

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-14 PhStat - Normal Probabilities  PHStat | Probability & Prob. Distributions | Normal…

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-15 Assessing Normality  Not all continuous random variables are normally distributed  It is important to evaluate how well the data set is approximated by a normal distribution

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-16 Assessing Normality  Construct charts or graphs  For small, or moderate sized data sets - do a stem-and- leaf display and box-and-whisker plot look symmetric?  For large data sets, does the histogram or polygon appear bell-shaped?  Compute descriptive summary measures  Do the mean, median and mode have similar values?  Is the interquartile range approximately 1.33  ?  Is the range approximately 6  ?  Evaluate normal probability plot (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-17 A normal probability plot for data from a normal distribution will be approximately linear: Z X The Normal Probability Plot (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-18 Normal Probability Plot Left-SkewedRight-Skewed Rectangular Z X (continued) Z X Z X Nonlinear plots indicate a deviation from normality

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-19 Normal Probability Plot in PHStat  PHStat | Probability & Prob. Distributions | Normal Probability Plot…  Need actual data

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-20 The Uniform Distribution Continuous Probability Distributions Normal Uniform Exponential

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-21 The Uniform Distribution  The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable (a die - P(X)=1/6)  Also called a rectangular distribution

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-22 The Continuous Uniform Distribution: where f(X) = value of the density function at any X value a = minimum value of X b = maximum value of X The Uniform Distribution (continued) ab X f(X) For a die

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-23 Uniform Distribution Example Example: Uniform Probability Distribution Over the range 2 ≤ X ≤ 6: f(X) = =.25 for 2 ≤ X ≤ X f(X)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-24 The Exponential Distribution Continuous Probability Distributions Normal Uniform Exponential

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-25 The Exponential Distribution  Used to model the length of time 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

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-26 Exponential Distributions  Describes time or distance between events is the inverse of the Poisson distribution  Density function   Parameters  f(X) X = 0.5 = 2.0

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-27 Exponential Distribution Example Example: Customers arrive at the service counter at the rate of 15 per hour. What is the probability that the arrival time between consecutive customers is less than three minutes?  The mean number of arrivals per hour is 15, so λ = 15  Three minutes is.05 hours  P(arrival time <.05) = 1 – e -λX = 1 – e -(15)(.05) =.5276  So there is a 52.76% probability that the arrival time between successive customers is less than three minutes

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-28 Exponential Distribution in PHStat  PHStat | Probability & Prob. Distributions | Exponential…

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-29 Sampling Distributions Sampling Distributions of the Mean Sampling Distributions of the Proportion

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-30 Sampling Distributions  A sampling distribution is a distribution of all of the possible values for the mean or proportion from a given size sample selected from a population

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-31 Developing a Sampling Distribution  Assume there is a population …  Population size N=4  Random variable, X, is age of individuals  Values of X: 18, 20, 22, 24 (years) A B C D

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap A B C D Uniform Distribution P(x) x (continued) Summary Measures for the Population Distribution: Developing a Sampling Distribution

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap possible samples (sampling with replacement) Now consider all possible samples of size n=2 (continued) Developing a Sampling Distribution 16 Sample Means

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-34 Sampling Distribution of All Sample Means P(X) X Sample Means Distribution 16 Sample Means _ Developing a Sampling Distribution (continued) (no longer uniform) _

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-35 Summary Measures of this Sampling Distribution: Developing a Sampling Distribution (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-36 Comparing the Population with its Sampling Distribution P(X) X A B C D Population N = 4 P(X) X _ Sample Means Distribution n = 2 _

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-37 Standard Error of the Mean  Different samples of the same size from the same population will yield different sample means  A measure of the variability from sample to sample is given by the Standard Error of the Mean:  Note that the standard error of the mean decreases as the sample size increases

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-38 If the Population is Normal  If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and (This assumes that sampling is with replacement or sampling is without replacement from an infinite population)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-39  For sampling with replacement: As n increases, decreases Sampling Distribution Properties Larger sample size Smaller sample size

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-40 Population and Sampling Distribution Population All samples of size n  Sampling Distribution

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-41 If the Population is not Normal  We can apply the Central Limit Theorem:  Even if the population is not normal,  …sample means from the population will be approximately normal as long as the sample size is large enough  …and the sampling distribution will have and

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-42 n↑n↑ Central Limit Theorem As the sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-43 Population Distribution Sampling Distribution (becomes normal as n increases) Central Tendency Variation (Sampling with replacement) Larger sample size Smaller sample size If the Population is not Normal (continued) Sampling distribution properties:

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-44 How Large is Large Enough?  For most distributions, n > 30 will give a sampling distribution that is nearly normal  For fairly symmetric distributions, n > 15  For normal population distributions, the sampling distribution of the mean is always normally distributed

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-45 Example  Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.  What is the probability that the sample mean is between 7.8 and 8.2?

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-46 Example Solution:  n > 30 … so the sampling distribution is approximately normal  … with mean = 8  …and standard deviation  Use these as parameters of the normal distribution and find the probability that the sample mean is between 7.8 and 8.2 (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-47 Sampling Distributions of the Proportion Sampling Distributions Sampling Distributions of the Mean Sampling Distributions of the Proportion

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-48 Population Proportions, p p = the proportion of the population having some characteristic Sample proportion ( p s ) provides an estimate of p:  0 ≤ p s ≤ 1  p s has a binomial distribution (assuming sampling with replacement from a finite population or without replacement from an infinite population)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-49 Sampling Distribution of p  Approximated by a normal distribution if:  where and (where p = population proportion) Sampling Distribution P( p s ) psps

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-50 Example  If the true proportion of voters who support Proposition A is p =.4, what is the probability that a sample of size 200 yields a sample proportion between.40 and.45?  i.e.: if p =.4 and n = 200, what is P(.40 ≤ p s ≤.45) ?

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-51 Example  if p =.4 and n = 200, what is P(.40 ≤ p s ≤.45) ? (continued) Find : Use PhStat Normal distribution with mean of.4 and standard deviation.03464

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-52 Sampling from Finite Populations  Modify standard error if sample size (n) is large relative to population size (N )   Use finite population correction factor (fpc)  Standard error with FPC  

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-53 Chapter Summary  Presented key continuous distributions  normal, uniform, exponential  Found probabilities using PhStat and Excel  Recognized when to apply different distributions  Applied distributions to decision problems

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-54 Chapter Summary  Introduced sampling distributions  Described the sampling distribution of the mean  For normal populations  Using the Central Limit Theorem  Described the sampling distribution of a proportion  Calculated probabilities using sampling distributions (continued)