Slide 1Test 4Practice ProblemsAdditional Reading and ExamplesSlide 5Motivating ExampleMotivating Example ProblemsMotivating Example AnswersMotivating Examples AnswersPractice ProblemsPractice Problem Answerst-Distributionst-Distributionst-DistributionsShapeCenterUnusual FeaturesSpreadSlide 19t-DistributionsExample 55Example 55Example 55Sampling DistributionsSampling DistributionSampling Distribution ExampleSampling Distribution ExampleSampling Distribution ExampleSampling Distribution ExampleSampling Distribution ExampleSTAT 210Lecture 22t-Distributions and Sampling DistributionsOctober 16, 2017Test 4Wednesday, October 18Covers chapter VI, pages 139 – 168Combination of multiple choice / short answer questions and problems.Formulas and tables provided, please bring calculator and writing instrument.Practice ProblemsPages 162 through 165Relevant problems: VI.16 and VI.17Recommended problems: VI.16 and VI.17Additional Reading and ExamplesRead pages 158 through 160TOP HAT 2Motivating ExampleThe population consists of a group of people in an investment club, and the variable of interest is X = the age of each person in the investment club.Suppose that the ages of people in the investment club follow an approximate normal distribution with mean 41 years and standard deviation 11 years.Then X ~ N(41, 11)Motivating Example ProblemsX ~ N(41, 11) 1. Find the probability that an investor’s age is less than 19. P(X < 19) = ???2. Find the probability that an investor’s age is greater than 57. P(X > 57) = ???3. What must an investor’s age be such that the probability of being less than that age is .1685?Motivating Example AnswersX ~ N(41, 11) 1. P(X < 19) = P(Z < 19 – 41) = P(Z < -2.00) = .0228 11Calculator: normalcdf (-1E99, 19, 41, 11)2. P(X > 57) = P(Z > 57 – 41) = P(Z > 1.45) 11 = 1 – P(Z < 1.45) = 1 - .9265 = .0735Calculator: normalcdf (57, 1E99, 41, 11)Motivating Examples AnswersX ~ N(41, 11)3. Find x such that P(X < x) = .1685 Find p = .1685 in the body of the table, read across and up to find z = -0.96. x = m + zs = 41 – 0.96(11) = 41 – 10.56 = 30.44 Calculator: invNorm (.1685, 41, 11)Practice ProblemsSuppose X ~ N(450, 85)1. Find the probability that X is greater than 500. P(X > 500) = ???2. Find the probability that X is between 400 and 640. P(400 < X < 640) = ???Practice Problem AnswersSuppose X ~ N(450, 85)1. Find the probability that X is greater than 500. P(X > 500) = P(Z > (500-450)/85) = P(Z > 0.59) = 1 – P(Z < 0.59) = 1-.7224 = .2776Calculator: normalcdf(500, 1E99, 450, 85)2. Find the probability that X is between 400 and 640. P(400 < X < 640) = P((400-450)/85 < Z < (640-450)/85) = P(-0.59 < Z < 2.24) = P(Z < 2.24) – P(Z < -0.59) = .9875 - .2776 = .7099Calculator: normalcdf(400, 640, 450, 85)t-DistributionsThe past three lectures we have discussed the normal distributions. To specify the normal distribution being used one must specify the values of the population mean m and the population standard deviation s.In many cases the value of the population standard deviation s is not known, and hence it is not possible to use the normal distributions. In these cases the t-distributions are often used instead of the standard normal (Z) distribution.t-DistributionsIf the population standard deviation s is unknown, then instead of using the standard normal distribution Z, we use a similar distribution called the Student’s t-distribution.t-DistributionsOn the next few slides we compare the standard normal (Z) distribution and the t-distributions.ShapeThe Z-distribution and all t-distributions have the same symmetric, bell-shape.CenterThe Z-distribution and all t-distributions have the same mean, which is 0.Mean = 0Unusual FeaturesThe Z-distribution and all t-distributions have the same symmetric, bell-shape, and therefore none of the distributions have any unusual features.SpreadThe standard deviation of the Z-distribution is 1, but the standard deviation of a t-distribution will be greater than or equal to 1 and depends on what is called the degrees of freedom (denoted df). As the degrees of freedom increases, the t-distribution gets closer and closer to the Z-distribution to the point that for df = ∞ the Z and t-distributions are the same and the standard deviation of the t-distribution with df = ∞ is 1 (the same as for Z).TOP HATt-DistributionsThe t-distribution is tabled on page 340.The degrees of freedom are located in the left-most column.The upper-tail probabilities are located in the top row.The t critical values are located in the body of the table.NOTE: The last row is labeled Z*, and corresponds to an infinite number of degrees of freedom, and implies that as the sample size (and hence the degrees of freedom) increasesthen the t-distribution approaches the Z distribution.Example 55a. df = 11 p = .05 t11 = 1.796Example 55b. df = 27 p = .01 t27 = 2.473Example 55c. df = 68 p = .05 Use df = 60 or df = 80 t60 = 1.671 t80 = 1.664Sampling DistributionsStatistical inference involves using data collected and statistics computed from a sample to make statements about some parameter of the population.Formulas used to make such statistical inferences are derived from the sampling distributions of the statistics being used.Sampling DistributionA sampling distribution of a statistic is the distribution of values taken by the statistic in a large number of simple random samples of the same size n taken from the same population.Sampling Distribution ExampleSuppose the population consists of all VCU students.The parameter of interest is m = the mean age of all VCU students.Do you know the mean age of all VCU students?Sampling Distribution ExampleSuppose the population consists of all VCU students.The parameter of interest is m = the mean age of all VCU students.Do you know the mean age of all VCU students?Assuming no, then it must be estimated using a statistical inference procedure.Sampling Distribution ExampleSuppose the population consists of all VCU students.The parameter of interest is m = the mean age of all VCU students.Suppose each person in this room selects a simple random sample of n = 100 VCU students and computes X = the mean age of the students in their sample.Sampling Distribution ExampleSuppose the population consists of all VCU students.The parameter of interest is m = the mean age of all VCU students.Student 1: sample of 100 students, X =
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