# VCU STAT 210 - Lecture22 (30 pages)

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## Lecture22

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- Pages:
- 30
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 22 t Distributions and Sampling Distributions October 16 2017 Test 4 Wednesday October 18 Covers chapter VI pages 139 168 Combination of multiple choice short answer questions and problems Formulas and tables provided please bring calculator and writing instrument Practice Problems Pages 162 through 165 Relevant problems VI 16 and VI 17 Recommended problems VI 16 and VI 17 Additional Reading and Examples Read pages 158 through 160 TOP HAT 2 Motivating Example The 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 Problems X 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 Answers X N 41 11 1 P X 19 P Z 19 41 P Z 2 00 0228 11 Calculator 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 0735 Calculator normalcdf 57 1E99 41 11 Motivating Examples Answers X 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 Problems Suppose 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 Answers Suppose 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 2776 Calculator 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 7099 Calculator normalcdf 400 640 450 85 t Distributions The 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 tdistributions are often used instead of the standard normal Z distribution t Distributions If 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 Distributions On the next few slides we compare the standard normal Z distribution and the t distributions Shape The Z distribution and all t distributions have the same symmetric bell shape Center The Z distribution and all t distributions have the same mean which is 0 Mean 0 Unusual Features The Z distribution and all t distributions have the same symmetric bell shape and therefore none of the distributions have any unusual features Spread The 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 HAT t Distributions The 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 increases then the t distribution approaches the Z distribution Example 55 a df 11 t11 1 796 p 05 Example 55 b df 27 t27 2 473 p 01 Example 55 c df 68 p 05 Use df 60 or df 80 t60 1 671 t80 1 664 Sampling Distributions Statistical 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 Distribution A 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 Example Suppose 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 Example Suppose 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 Example Suppose 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 Example Suppose 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 23 8 Student 2 sample of 100 students X 24 3 Student 3 sample of 100 students X 22 7 Sampling Distribution Example Suppose the population consists of all VCU students The parameter of interest is m the mean age of all VCU students These sample mean values 23 8 24 3 22 7 25 0 are now considered a data set We could construct a stem and leaf plot a histogram or a boxplot for this data This plot would show the sampling distribution of

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