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  1. OMS 201 Review

  2. OMS 201 Review Range • The range of a data set is the difference between the largest and smallest data values. • It is the simplest measure of dispersion. • It is very sensitive to the smallest and largest data values.

  3. OMS 201 Review Variance The variance is the average of the squared n differences between each data value and the mean. If the data set is a sample, the variance is denoted n - 2 - å by . s 2 ( ) x x i i 2 = = s - - 1 n If the data set is a population, the variance is n s denoted by . 2 2 - m å ) ( x ) 2 s = N

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  6. OMS 201 Review Random Variables • A random variableis a numerical description of the outcome of an experiment. • A discrete random variablemay assume either a finite number of values or an infinite sequence of values. • A continuous random variablemay assume any numerical value in an interval or collection of intervals.

  7. OMS 201 Review Expected Value and Variance • The expected value, or mean, of a random variable is a measure of its central location. • Expected value of a discrete random variable: E (x ) =  = x f (x )

  8. n ! - x ( n x ) = - f ( x ) p ( 1 p ) - x ! ( n x ) ! OMS 201 Review The Binomial Probability Distribution where f (x ) = the probability of x successes in n trials n = the number of trials p = the probability of success on any one trial

  9. OMS 201 Review Example: Evans Electronics • Using the Tables of Binomial Probabilities

  10. f (x ) x  OMS 201 Review The Normal Probability Distribution • Graph of the Normal Probability Density Function

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  12. OMS 201 Review Sampling Distribution of for the SAT Scores From Census From Census

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  14. OMS 201 Review Using the standard normal probability table with z = 10/11.3 = .88, we have area = (.3106)(2) = .6212. There is a .6212 probability that the sample mean will be within +/-10 of the actual population mean. How do we increase this probability? When n is increased the standard error of the mean is decreased Large sample sizes will provide a higher probability that the sample mean is within a specified distance of the mean!

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  19. OMS 201 Review Interval Estimation of a Population Mean:Small-Sample Case (n < 30) • If Population is not normally distributed: • The only option is to increase the sample size to n> 30 and use the large-sample interval-estimation procedures. • If Population is normally distributed and  is known: • The large-sample interval-estimation procedure can be used. • If Population is normally distributed and  is unknown: • The appropriate interval estimate is based on a probability distribution known as the t distribution.

  20. OMS 201 Review t Distribution • The t distribution is a family of similar probability distributions. • A specific t distribution depends on a parameter known as the degrees of freedom. • As the number of degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller. • A t distribution with more degrees of freedom has less dispersion. • The mean of the t distribution is zero.

  21. OMS 201 Review Interval Estimation of a Population Mean:Small-Sample Case (n < 30) with  Unknown • The interval estimate is given by: where 1 - is the confidence coefficient t/2is the t value providing an area of /2 in the upper tail of a t distribution with n - 1 degrees of freedom s is the sample standard deviation

  22. OMS 201 Review Interval Estimationof a Population Proportion • The interval estimate is given by: where 1 - is the confidence coefficient z/2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution is the sample proportion

  23. OMS 201 Review Goodness of Fit Test Some forecasting procedures assume that the sample data fit a known distribution. Ho: The sample data fit the distribution Ha: The sample data do not fit the distribution Table value with 3 (4-1) degrees of freedom = 7.81473 Do Not Reject Ho: Calculate a chi-square statistic. Compare to chi-square table with k-1 degrees of freedom, where k = # of categories. If the calculated value is less than the table value, it is a good fit (Do not reject Ho:). If the calculated value is larger than the table value, a poor fit is indicated (Reject Ho:).