hypothesis testing 3.pptx technics and testing

Hypotheses technologies and its testing

1. Hypothesis Testing; Z-Test, T-Test, F-Test

2. What is Hypothesis?  Hypothesis is a predictive statement, capable of being tested by scientific methods, that relates an independent variables to some dependent variable.  A hypothesis states what we are looking for and it is a proportion which can be put to a test to determine its validity e.g. Students who receive counseling will show a greater increase in creativity than students not receiving counseling 3

3. Characteristics of Hypothesis  Clear and precise.  Capable of being tested.  Stated relationship between variables.  limited in scope and must be specific.  Stated as far as possible in most simple terms so that the same is easily understand by all concerned. But one must remember that simplicity of hypothesis has nothing to do with its significance.  Consistent with most known facts.  Responsive to testing with in a reasonable time. One can’t spend a life time collecting data to test it.  Explain what it claims to explain; it should have empirical reference. 4

4. Null Hypothesis • The Null hypothesis H represents a theory that has been put forward either because it is believed to be true. • it is used as a basis for an argument and has not been proven. • For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is no better, on average, than the current drug. We would write H : there is no difference between the two drugs on an average.

5. Null Hypothesis 5  It is an assertion that we hold as true unless we have sufficient statistical evidence to conclude otherwise.  Null Hypothesis is denoted by 𝐻  If a population mean is equal to hypothesised mean then Null Hypothesis can be written as 𝐻 : 𝜇 = 𝜇

6. Alternative Hypothesis • • • The alternative hypothesis, H A , is a statement of what a statistical hypothesis test is set up to establish. For example, in the clinical trial of a new drug, the alternative hypothesis might be that the new drug has a different effect, on average, compared to that of the current drug. We would write H A : the two drugs have different effects, on average. or H A : the new drug is better than the current drug, on average. The result of a hypothesis test: ‘Reject H in favor of H A ’ OR ‘Do not reject H ’

7. Alternative Hypothesis  The Alternative hypothesis is negation of null hypothesis and is denoted by 𝐻 𝑎 If Null is given as 𝐻 : 𝜇 = 𝜇 Then alternative Hypothesis can be written as 𝐻 𝑎 : 𝜇 ≠ 𝜇 𝐻 𝑎 : 𝜇 > 𝜇 𝐻 𝑎 : 𝜇 < 𝜇 6

8. Level of significance and confiden c e  Significance means the percentage risk to reject a null hypothesis when it is true and it is denoted by 𝛼 . Generally taken as 1%, 5%, 10%  (1 − 𝛼) is the confidence interval in which the null hypothesis will exist when it is true. 7

9. Risk of rejecting a Null Hypothesis 8 when it is true Desi g nati o n Risk 𝜶 Confidence 𝟏 − 𝜶 Description Sup e rcr i t i c a l 0.001 0.1% 0.999 99.9% More than $100 million (Large loss of life, e.g. nuclear disaster Critical 0.01 1% 0.99 99% Less than $100 million (A few lives lost) Important 0.05 5% 0.95 95% Less than $100 thousand (No lives lost, injuries occur) Moderate 0.10 10% 0.90 90% Less than $500 (No injuries occur)

10. Type I and Type II Error Situation Decision Accept Null Reject Null Null is true Correct Type I error ( 𝛼 𝑒𝑟𝑟𝑜𝑟 ) Null is false Type II error ( 𝛽 𝑒𝑟𝑟𝑜𝑟 ) Correct 9

11. Type I and Type II Errors 1. Type I error refers to the situation when we reject the null hypothesis when it is true (H is wrongly rejected). e.g H : there is no difference between the two drugs on average. Type I error will occur if we conclude that the two drugs produce different effects when actually there isn’t a difference. Prob (Type I error) = significance level = α 2. Type II error refers to the situation when we accept the null hypothesis when it is false. H : there is no difference between the two drugs on average. Type II error will occur if we conclude that the two drugs produce the same effect when actually there is a difference. Prob (Type II error) = ß

12. Type I Error A type I error , also known as an error of the first kind It occurs when the null hypothesis ( H ) is true, but is rejected. The rate of the type I error is called the size of the test. It is denoted by the Greek letter α (alpha). It usually equals the significance level of a test . If type I error is fixed at 5 %, it means that there are about 5% chances in 100% that we will reject H when H is true.

13. Type II Error Type II error , also known as an error of the second kind It occurs when the null hypothesis is false, but due to error fails to be rejected. Type II error means accepting the hypothesis which should have been rejected . A Type II error is committed when we fail to believe a truth. T h e r ate of the t y pe II error is d e no t e d by t he G r eek l e t ter β (beta) and related to the power of a test (which equals 1-β ) .

14. In the tabular form two error can be presented as follows -

15. Two tailed test @ 5% Significance level Acceptance and Rejection regions in case of a Two tailed test 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑔𝑖𝑜𝑛 /𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (𝛼 = 0.025 𝑜𝑟 2.5%) 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑔𝑖𝑜𝑛 /𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (𝛼 = 0.025 𝑜𝑟 2.5%) Suitable When 𝐻 : 𝜇 = 𝜇 𝐻 𝑎 : 𝜇 ≠ 𝜇 𝑇𝑜𝑡𝑎𝑙 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 𝑟𝑒𝑔𝑖𝑜𝑛 𝑜𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (1 − 𝛼) = 95% 𝐻 : 𝜇 = 𝜇 10

16. Left tailed test @ 5% Significance level Acceptance and Rejection regions in case of a left tailed test 𝐻 : 𝜇 = 𝜇 𝑇𝑜𝑡𝑎𝑙 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 𝑟𝑒𝑔𝑖𝑜𝑛 𝑜𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (1 − 𝛼) = 95% 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑔𝑖𝑜𝑛 /𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (𝛼 = 0.05 𝑜𝑟 5%) Suitable When 𝐻 : 𝜇 = 𝜇 𝐻 𝑎 : 𝜇 < 𝜇 11

17. Right tailed test @ 5% Significance level Acceptance and Rejection regions in case of a Right tailed test Suitable When 𝐻 : 𝜇 = 𝜇 𝐻 𝑎 : 𝜇 > 𝜇 𝑇𝑜𝑡𝑎𝑙 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 𝑟𝑒𝑔𝑖𝑜𝑛 𝑜𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (1 − 𝛼) = 95% 𝐻 : 𝜇 = 𝜇 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑔𝑖𝑜𝑛 /𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (𝛼 = 0.05 𝑜𝑟 5%) 12

18. Procedure for Hypothesis Te s t i ng State the null (Ho)and alternate (Ha) Hypothesis State a significance level; 1%, 5%, 10% etc. Decide a test statistics; z-test, t- test, F-test. Calculate the value of test statistics Calculate the p- value at given significance level from the table Compare the p-value with calculated value P-value > Ca l cu l a t ed value P-value < Ca l cu l a t ed value Accept Ho Reject Ho 13

19. Hy p othesis Testing of Means Z-TEST AND T-TEST 14

20. Z-Test for testing means Test Condition  Population normal and infinite  Sample size large or small,  Population variance is known  Ha may be one-sided or two sided Test Statistics 𝑋 −𝜇 𝐻 𝑧 = 𝜎 𝑝 𝑛 15

21. Z-Test for testing means Test Condition  Population normal and finite,  Sample size large or small,  Population variance is known  Ha may be one-sided or two sided Test Statistics 𝑋 − 𝜇 𝐻 𝑧 = 𝜎 𝑝 𝑛 × 𝑁 − 𝑛 𝑁 − 1 16

22. Z-Test for testing means Test Condition  Population is infinite and may not be normal,  Sample size is large,  Population variance is unknown  Ha may be one-sided or two sided Test Statistics 𝑋 −𝜇 𝐻 𝑧 = 𝜎 𝑠 𝑛 17

23. Z-Test for testing means Test Condition  Population is finite and may not be normal,  Sample size is large,  Population variance is unknown  Ha may be one-sided or two sided Test Statistics 𝑋 − 𝜇 𝐻 𝑧 = 𝜎 𝑠 𝑛 × 𝑁 − 𝑛 𝑁 − 1 18

24. T-Test for testing means Test Condition  Population is infinite and normal,  Sample size is small,  Population variance is unknown  Ha may be one-sided or two sided Test Statistics 𝑋 −𝜇 𝐻 𝑡 = 𝜎 𝑠 𝑛 𝑤𝑖𝑡ℎ 𝑑. 𝑓. = 𝑛 − 1 𝜎 𝑠 = 𝑋 𝑖 − 𝑋 2 (𝑛 − 1) 19

25. T-Test for testing means Test Condition  Population is finite and normal,  Sample size is small,  Population variance is unknown  Ha may be one-sided or two sided Test Statistics 𝑤𝑖𝑡ℎ 𝑑. 𝑓. = 𝑛 − 1 𝜎 𝑠 = 𝑋 𝑖 − 𝑋 2 (𝑛 − 1) 𝑋 − 𝜇 𝐻 𝑡 = 𝜎 𝑠 𝑛 × 𝑁 − 𝑛 𝑁 − 1 20

26. Hypo t he s is testing for difference between means Z-TEST, T-TEST 21

27. Z-Test for testing difference between means Test Condition  Populations are normal  Samples happen to be large,  Population variances are known  Ha may be one-sided or two sided Test Statistics 𝑋 − 𝑋 𝑧 = 1 2 𝑝1 + 𝑝2 𝜎 2 𝜎 2 𝑛 1 𝑛 2 22

28. Z-Test for testing difference between means Test Condition  Populations are normal  Samples happen to be large,  Presumed to have been drawn from the same population  Population variances are known  Ha may be one-sided or two sided Test Statistics 𝑧 = 𝑋 1 − 𝑋 2 𝑝 𝜎 2 + 1 1 𝑛 1 𝑛 2 23

29. T-Test for testing difference between means Test Condition  Samples happen to be small,  Presumed to have been drawn from the same population  Population variances are unknown but assumed to be equal  Ha may be one-sided or two sided Test Statistics 𝑡 = 𝑋 1 − 𝑋 2 𝑠 1 𝑛 1 − 1 𝜎 2 + 𝑛 2 − 1 𝜎 2 𝑛 1 + 𝑛 2 − 2 𝑠2 × + 1 1 𝑛 1 𝑛 2 𝑤𝑖𝑡ℎ 𝑑. 𝑓. = (𝑛 1 + 𝑛 2 − 2) 24

30. Hypothesis Testing for comparing two related samples PAIRED T-TEST 25

31. Paired T-Test for comparing two related samples Test Condition  Samples happens to be small  Variances of the two populations need not be equal  Populations are normal  Ha may be one sided or two sided Test Statistics 𝐷 − 𝑡 = 𝜎 𝑑𝑖𝑓𝑓. 𝑛 𝑤𝑖𝑡ℎ (𝑛 − 1) 𝑑. 𝑓. 𝐷 = Mean of differences 𝜎 𝑑𝑖𝑓𝑓. = Standard deviation of differences 𝑛 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑎𝑡𝑐ℎ𝑒𝑑 𝑝𝑎𝑖𝑟𝑠 26

32. Hypothesis Testing of propor t ions Z- TEST 27

33. Z-test for testing of proportions Test Condition  Use in case of qualitative data  Sampling distribution may take the form of binomial probability distribution  Ha may be one sided or two sided  𝑀𝑒𝑎𝑛 = 𝑛. 𝑝  𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 𝑛. 𝑝. 𝑞 Test statistics 𝑧 = 𝑝 − 𝑝 𝑝. 𝑞 𝑛 𝑝 = 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑢𝑐𝑒𝑠𝑠 28

34. Hypothesis Testing for difference between pro p or t ions Z- TEST 29

35. Z-test for testing difference between proportions Test Condition  Sample drawn from two different populations  Test confirm, whether the difference between the proportion of success is significant  Ha may be one sided or two sided Test statistics 𝑧 = 𝑝 1 − 𝑝 2 𝑝 1 𝑞 1 + 𝑝 2 𝑞 2 𝑛 1 𝑛 2 𝑝 1 = proportion of success in sample one 𝑝 2 = proportion of success in sample two 30

36. Hypothesis testing of equality of variances of two normal populations F - TEST 31

37. F-Test for testing equality of variances of two normal populations Test conditions  The populations are normal  Samples have been drawn randomly  Observations are independent; and  There is no measurement error  Ha may be one sided or two sided Test statistics 𝐹 = 𝑠1 𝜎 2 𝑠 2 𝜎 2 𝑤𝑖𝑡ℎ 𝑛 1 − 1 and 𝑛 2 − 1 d. f. 𝑠 1 𝜎 2 𝑝 1 is the sample estimate for 𝜎 2 𝑠 2 𝜎 2 𝑝 2 is the sample estimate for 𝜎 2 32

38. Limitations of the test of Hypothesis  Testing of hypothesis is not decision making itself; but help for decision making  Test does not explain the reasons as why the difference exist, it only indicate that the difference is due to fluctuations of sampling or because of other reasons but the tests do not tell about the reason causing the difference.  Tests are based on the probabilities and as such cannot be expressed with full certainty.  Statistical inferences based on the significance tests cannot be said to be entirely correct evidences concerning the truth of the hypothesis. 33

39. Thank You 34