Continuous Probability Distributions Continuous random variable –Values from interval of numbers –Absence of gaps Continuous probability distribution –Distribution of continuous random variable Most important continuous probability distribution –The normal distribution

The Uniform Distribution “Rectangular shaped” Every value between a and b is equally likely The mean and median are in the middle Prob(X<=v) is the area on the left of v Mean Median X f(X)  ba v

The Normal Distribution “Bell shaped” Symmetrical Mean, median and mode are equal Interquartile range equals 1.33  % rule Random variable has infinite range Mean Median X f(X) 

The Mathematical Model

Many Normal Distributions By varying the parameters  and , we obtain different normal distributions There are an infinite number of normal distributions

Finding Probabilities Probability is the area under the curve! c d X f(X)f(X)

Which Table to Use? An infinite number of normal distributions means an infinite number of tables to look up!

Solution: The Cumulative Standardized Normal Distribution Z Cumulative Standardized Normal Distribution Table (Portion) Probabilities Shaded Area Exaggerated Only One Table is Needed Z = 0.12

Standardizing Example Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

Example: Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = 0.21 Example: (continued)

Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = Example: (continued)

Example: Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

Example: (continued) Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = 0.30

.6217 Finding Z Values for Known Probabilities Z Cumulative Standardized Normal Distribution Table (Portion) What is Z Given Probability = ? Shaded Area Exaggerated.6217

Recovering X Values for Known Probabilities Normal Distribution Standardized Normal Distribution

Finding Probabilities for X Values Using Excel Excel function: =NORMDIST(x,mean,standard_deviation,TRUE) =NORMSDIST(z,TRUE) Example Prob.(weight <= 165 lbs) when mean=180, std_dev=20: =NORMDIST(165,180,20,true) Answer: Prob.(weight >= 185 lbs) ? Prob.(weight <= 165 and weight <= 185 lbs) ?

Finding X Values for Known Probabilities Using Excel Excel function: =NORMINV(probabiltiy,mean,standard_deviation) =NORMSINV(probability) Example Prob.(weight <= X)= 0.2(mean=180, std_dev=20) =NORMINV(0.2,180,20) Answer: X=163 Prob.(weight >= X)=0.4 X? Answer: X=185

Generating Random Variables Using Excel Excel can be used to generate Discrete and Continuous Random Variables Complex Probabilistic Models can be constructed and simulation can give insight and suggest managerial decisions Tutorial

Assessing Normality Not all continuous random variables are normally distributed It is important to evaluate how well the data set seems to be adequately approximated by a normal distribution

Assessing Normality Construct charts –For large data sets, does the histogram appear bell- shaped? Compute descriptive summary measures –Do the mean, median and mode have similar values? –Is the interquartile range approximately 1.33  ? –Does the data obey the percent rule? –Is the range approximately 6  ? (continued)

Assessing Normality Observe the distribution of the data set –Do approximately 2/3 of the observations lie between mean 1 standard deviation? –Do approximately 4/5 of the observations lie between mean 1.28 standard deviations? –Do approximately 19/20 of the observations lie between mean 2 standard deviations? (continued)

Why Study Sampling Distributions Sample statistics are used to estimate population parameters –e.g.: Estimates the population mean Problems: different samples provide different estimate –Large samples gives better estimate; Large samples costs more –How good is the estimate? Approach to solution: theoretical basis is sampling distribution

Sampling Distribution Theoretical probability distribution of a sample statistic Sample statistic is a random variable –Sample mean, sample proportion Results from taking all possible samples of the same size

Example Population: 100 subjects, numbered from 1 to 100 Take sample of 10 and compute average Take another sample, etc. Excel workbookExcel

Developing Sampling Distributions Assume there is a population … Population size N=4 Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 measured in years A B C D

A B C D (18) (20) (22) (24) Uniform Distribution P(X) X Developing Sampling Distributions (continued) Summary Measures for the Population Distribution

All Possible Samples of Size n=2 16 Samples Taken with Replacement 16 Sample Means Developing Sampling Distributions (continued)

Sampling Distribution of All Sample Means P(X) X Sample Means Distribution 16 Sample Means _ Developing Sampling Distributions (continued)

Summary Measures of Sampling Distribution Developing Sampling Distributions (continued)

Comparing the Population with its Sampling Distribution P(X) X Sample Means Distribution n = 2 A B C D (18) (20) (22) (24) Population N = 4 P(X) X _

Properties of Summary Measures –i.e. is unbiased Standard error (standard deviation) of the sampling distribution is less than the standard error of other unbiased estimators For sampling with replacement: –As n increases, decreases

Unbiasedness BiasedUnbiased P(X)

Less Variability Sampling Distribution of Median Sampling Distribution of Mean P(X)

Effect of Large Sample Larger sample size Smaller sample size P(X)

When the Population is Normal Central Tendency Variation Sampling with Replacement Population Distribution Sampling Distributions

When the Population is Not Normal Central Tendency Variation Sampling with Replacement Population Distribution Sampling Distributions

Central Limit Theorem As sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population

How Large is Large Enough? For most distributions, n>30 For fairly symmetric distributions, n>15 For normal distribution, the sampling distribution of the mean is always normally distributed

Example: Sampling Distribution Standardized Normal Distribution

Population Proportions Categorical variable –e.g.: Gender, voted for Bush, college degree Proportion of population having a characteristic Sample proportion provides an estimate – If two outcomes, X has a binomial distribution –Possess or do not possess characteristic

Sampling Distribution of Sample Proportion Approximated by normal distribution – –Mean: –Standard error: p = population proportion Sampling Distribution P(p s ) psps

Standardizing Sampling Distribution of Proportion Sampling Distribution Standardized Normal Distribution

Example: Sampling Distribution Standardized Normal Distribution