# VCU STAT 210 - Lecture25 (86 pages)

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## Lecture25

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- Pages:
- 86
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 25 Tests of Significance October 25 2017 Practice Problems Pages 187 and 188 Relevant problems VII 4 and VII 5 Recommended problems VII 4 and VII 5 Additional Reading and Examples Read pages 185 and 186 Motivating Example Suppose one is interested in determining the mean IQ of all students at this university What is the population of interest Population of interest consists of all VCU students What is the parameter of interest Parameter of interest is m the mean IQ of all VCU students Top Hat Tests of Significance With statistical tests we conjecture that the unknown population parameter equals some value referred to as a statistical hypothesis and then we use the data in the sample to test whether this value is reasonable or not Motivating Example Suppose one is interested in determining the mean IQ of all students at this university Population of interest consists of all VCU students and the parameter of interest is m the mean IQ of all VCU students Suppose President Rao believes the mean IQ of all VCU students is 110 Hence the hypothesis conjecture is that m the mean IQ of all VCU students is 110 and data collected from the sample is used to test whether or not this conjecture is correct or not Tests of Signicance 1 The null hypothesis denoted by H0 is a conjecture about a population parameter that is presumed to be true It is usually a statement of no effect or no change Motivating Example Suppose President Rao believes the mean IQ of all VCU students is 110 Then the null hypothesis is H0 m 110 General Terms and Characteristics 2 The alternative or research hypothesis denoted by Ha or H1 is a conjecture about a population parameter that the researcher suspects or hopes is true Motivating Example Suppose President Rao believes the mean IQ of all VCU students is 110 Then the null hypothesis is H0 m 110 Suppose with the growth in the number of applicants to the university the university is now more selective in which students it admits and hence the belief is that the mean IQ of all VCU students is higher than in the past Then the alternative hypothesis is Ha m 110 Top Hat Tests of Significance 3 The test statistic is some quantity calculated from the sample data that we have collected It is used to determine the strength of the evidence against the null hypothesis Tests of Significance 3 The test statistic is some quantity calculated from the sample data that we have collected It is used to determine the strength of the evidence against the null hypothesis After the null and alternative hypotheses are stated a sample is selected from the population and data is collected on the individuals in this sample From this data a statistic can be computed This statistic is the test statistic Motivating Example Test H0 m 110 versus Ha m 110 We could select a random sample of 50 VCU students and determine the IQ of each student in the sample These 50 IQ values could then be added and the sum divided by 50 to produce the sample mean X In this case the sample mean X could be the test statistic Tests of Significance 3 The test statistic is some quantity calculated from the sample data that we have collected It is used to determine the strength of the evidence against the null hypothesis If the test statistic value calculated is close to the hypothesized value then it is likely that the null hypothesis is correct Motivating Example Test H0 m 110 versus Ha m 110 We could select a random sample of 50 VCU students and determine the IQ of each student in the sample These 50 IQ values could then be added and the sum divided by 50 to produce the sample mean X In this case the sample mean X could be the test statistic If the sample mean X is close to 110 say 111 8 then is likely the case that the null hypothesis is correct Tests of Significance 3 The test statistic is some quantity calculated from the sample data that we have collected It is used to determine the strength of the evidence against the null hypothesis If the test statistic value calculated is not close to the hypothesized value then it is likely that the null hypothesis is not correct and we can conclude that the alternative hypothesis is correct Motivating Example Test H0 m 110 versus Ha m 110 We could select a random sample of 50 VCU students and determine the IQ of each student in the sample These 50 IQ values could then be averaged to produce the sample mean X If the sample mean X is far above 110 say 125 then is likely the case that the null hypothesis is not correct and we conclude that the alternative hypothesis is correct Tests of Significance 4 When we carry out the test we assume the null hypothesis is true Hence the test will result in one of two decisions i Reject H0 Hence we have sufficient evidence to conclude that the alternative hypothesis is true Such a test is said to be significant Motivating Example Test H0 m 110 versus Ha m 110 Suppose from the sample of 50 VCU students the mean IQ is X 125 Since 125 is far above 110 then we would likely reject the null hypothesis H0 m 110 and hence conclude that the alternative hypothesis that the mean IQ of all VCU students is greater than 110 is correct conclude Ha m 110 is correct Tests of Significance 4 When we carry out the test we assume the null hypothesis is true Hence the test will result in one of two decisions i Reject H0 Hence we have sufficient evidence to conclude that the alternative hypothesis is true Such a test is said to be significant ii Fail to reject H0 Hence we do not have sufficient evidence to conclude that the alternative hypothesis is true Such a test is said to be insignificant Motivating Example Test H0 m 110 versus Ha m 110 Suppose from the sample of 50 VCU students the mean IQ is X 112 Since 112 is very close to 110 then we would likely fail to reject the null hypothesis H0 m 110 and hence cannot conclude that the alternative hypothesis that the mean IQ of all VCU students is greater than 110 is correct cannot conclude Ha m 110 is correct Tests of Significance i Reject H0 Hence we have sufficient evidence to conclude that the alternative hypothesis is true Such a test is said to be significant ii Fail to reject H0 Hence we do not have sufficient evidence to conclude that the alternative hypothesis is true Such a test is said to be insignificant The question becomes what is the cut off value for rejecting the null hypothesis versus failing to reject the null hypothesis Tests of Significance The null hypothesis is either correct or it

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