Hyp A

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Hyp A

  1. 1. Basic Business Statistics (8 th Module) Chapter 9 Fundamentals of Hypothesis Testing: One-Sample Tests
  2. 2. Chapter Topics <ul><li>Hypothesis Testing Methodology </li></ul><ul><li>Z Test for the Mean ( Known) </li></ul><ul><li>p-Value Approach to Hypothesis Testing </li></ul><ul><li>Connection to Confidence Interval Estimation </li></ul><ul><li>One-Tail Tests </li></ul><ul><li>t Test for the Mean ( Unknown) </li></ul><ul><li>Z Test for the Proportion </li></ul><ul><li>Potential Hypothesis-Testing Pitfalls and Ethical Issues </li></ul>
  3. 3. What is a Hypothesis? <ul><li>A Hypothesis is a Claim (Assumption) about the Population Parameter </li></ul><ul><ul><li>Examples of parameters are population mean or proportion </li></ul></ul><ul><ul><li>The parameter must be identified before analysis </li></ul></ul>I claim the mean GPA of this class is 3.5! © 1984-1994 T/Maker Co.
  4. 4. Testing of hypothesis <ul><li>The process of accepting rejecting a statistical hypothesis based on sampling is called testing of hypothesis. </li></ul><ul><li>Eg: The Hypothesis 5% of people in a city A-Are suffering from TB is a Statistical Hypothesis. </li></ul><ul><li>Taking a random sample of the size n from the population of the city, calculating the proportion of the individuals suffering from TB and deciding whether we have to accept or reject the hypothesis is called Statistical Test . </li></ul>
  5. 5. The Null Hypothesis, H 0 <ul><li>States the Assumption (Numerical) to be Tested </li></ul><ul><ul><li>E.g., The mean GPA is 3.5 </li></ul></ul><ul><li>Null Hypothesis is Always about a Population Parameter ( ), Not about a Sample Statistic ( ) </li></ul><ul><li>Is the Hypothesis a Researcher Tries to Reject </li></ul>
  6. 6. The Null Hypothesis, H 0 <ul><li>Begin with the Assumption that the Null Hypothesis is True </li></ul><ul><ul><li>Similar to the notion of innocent until proven guilty </li></ul></ul><ul><li>Refer to the Status Quo </li></ul><ul><li>Always Contains the “=” Sign </li></ul><ul><li>The Null Hypothesis May or May Not be Rejected </li></ul>(continued)
  7. 7. The Alternative Hypothesis, H 1 <ul><li>Is the Opposite of the Null Hypothesis </li></ul><ul><ul><li>E.g., The mean GPA is NOT 3.5 ( ) </li></ul></ul><ul><li>Challenges the Status Quo </li></ul><ul><li>Never Contains the “=” Sign </li></ul><ul><li>The Alternative Hypothesis May or May Not Be Accepted (i.e., The Null Hypothesis May or May Not Be Rejected) </li></ul><ul><li>Is Generally the Hypothesis that the Researcher Claims </li></ul>
  8. 8. Error in Making Decisions <ul><li>Type I Error </li></ul><ul><ul><li>Reject a true null hypothesis </li></ul></ul><ul><ul><ul><li>When the null hypothesis is rejected, we can say that “We have shown the null hypothesis to be false (with some ‘slight’ probability, i.e. , of making a wrong decision) </li></ul></ul></ul><ul><ul><li>Has serious consequences </li></ul></ul><ul><ul><li>Probability of Type I Error is </li></ul></ul><ul><ul><ul><li>Called level of significance </li></ul></ul></ul><ul><ul><ul><li>Set by researcher </li></ul></ul></ul>
  9. 9. Type 1 Error (  ) <ul><li>The size of a test is the probability of rejecting null hypothesis when it is true.It is denoted by and this type of error is called Type 1 error </li></ul><ul><li>  =P(reject H 0/ H 0 ) </li></ul><ul><li>Rejecting H0 , when H0 is True (Rejecting a True Hypothesis). Eg; </li></ul>
  10. 10. Type 2 Error(  ) <ul><li>Accepting H0 when H0 is False(Accepting a wrong Hypothesis). </li></ul> = Accept(H0/H1)
  11. 11. Error in Making Decisions <ul><li>Type II Error </li></ul><ul><ul><li>Fail to reject a false null hypothesis </li></ul></ul><ul><ul><li>Probability of Type II Error is </li></ul></ul><ul><ul><li>The power of the test is </li></ul></ul><ul><li>Probability of Not Making Type I Error </li></ul><ul><ul><li>Called the Confidence Coefficient </li></ul></ul>(continued)
  12. 12. Power of a Test <ul><li>Consider the case when the Hypothesis H 0 is not True, then there is a possibility of H 1 is True. </li></ul><ul><li>Power of a test is defined as the probability of rejecting the null hypothesis when it is actually false ie when H 1 is true. </li></ul><ul><ul><ul><ul><li>Power=P(reject H0/H1) </li></ul></ul></ul></ul><ul><li>=1 – P(Accept H 0 /H 1 ) </li></ul><ul><li>= I – PROBABILITY OF TYPE 11 ERROR </li></ul><ul><li>= I -  </li></ul>
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  14. 14. Standard Error <ul><li>The standard deviation of a statistic is called standard error of statistic. </li></ul><ul><li>Let X1,X2 etc be the means of possible samples of size n drawn from a population, the standard deviation of the sample ,means the standard error of these means. This is found to be  /  n, where  is the population standard deviation and n is the sample size. </li></ul>
  15. 15. Degrees of Freedom <ul><li>The number of degrees of freedom in a problem, distribution, etc., is the number of parameters which may be independently varied . It is the no of independent observations in a set’ </li></ul><ul><li>If we are choosing any 5 nos. , whose total is 100, we can exercise our independent choice for any 4 nos. Only the 5 th no. is fixed by virtue of the total being 100. In this example there is only 1 restriction in our freedom. Therefore the degrees of freedom is 4. </li></ul>
  16. 16. Significance Level <ul><li>The significance level of a statistical hypothesis test is a fixed probability of wrongly rejecting the null hypothesis H 0 , if it is in fact true. </li></ul>
  17. 17. Statistic <ul><li>A statistic (singular) is the result of applying a statistical algorithm to a set of data . In the calculation of the arithmetic mean , for example, the algorithm directs us to sum all the data values and divide by the number of data items. In this case, we call the mean a statistic. </li></ul>
  18. 18. Test Statistic <ul><li>A test statistic is a quantity calculated from our sample of data. Its value is used to decide whether or not the null hypothesis should be rejected in our hypothesis test. </li></ul>
  19. 19. Critical Value(s) <ul><li>The critical value(s) for a hypothesis test is a threshold to which the value of the test statistic in a sample is compared to determine whether or not the null hypothesis is rejected. </li></ul>
  20. 20. Critical Region <ul><li>The critical region CR , or rejection region RR , is a set of values of the test statistic for which the null hypothesis is rejected in a hypothesis test; that is, the sample space for the test statistic is partitioned into two regions; one region (the critical region) will lead us to reject the null hypothesis H 0 ', the other not. So, if the observed value of the test statistic is a member of the critical region, we conclude 'reject H 0 '; if it is not a member of the critical region then we conclude 'do not reject H 0 . </li></ul>
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