Overview

1. Definition Hypothesis in statistics, is a claim or statement about a property of a population Overview

2. Statement about value of population parameter Must contain condition of equality =, , or Test the Null Hypothesis directly RejectH0 or fail to rejectH0 Null Hypothesis: H0

3. Must be true if H0 is false , <, > ‘opposite’ of Null Alternative Hypothesis: Ha

4. If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis. Note about Forming Your Own Claims (Hypotheses)

5. HO The defendant Claim about a is not guilty population parameter HA The defendant Opposing claim about a is guilty population parameter Result The evidence The statistic indicates a convinces the rejection of HO, and the jury to reject alternate hypothesis is the assumption accepted. of innocence. The verdict is guilty Legal Trial Hypothesis Test

6. a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis Test Statistic

7. a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis For large samples, testing claims about population means Test Statistic x - µx z=  n

8. Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region

9. Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region Critical Region

10. Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region Critical Region

11. Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region Critical Regions

12. denoted by  the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. common choices are 0.05, 0.01, and 0.10 Significance Level

13. Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis Critical Value

14. Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis Critical Value Critical Value ( z score )

15. Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis Critical Value Reject H0 Fail to reject H0 Critical Value ( z score )

16. Two-tailed,Right-tailed,Left-tailed Tests The tails in a distribution are the extreme regions bounded by critical values.

17. H0: µ = 100 Ha: µ  100 Two-tailed Test

18. H0: µ = 100 Ha: µ  100 Two-tailed Test  is divided equally between the two tails of the critical region

19. H0: µ = 100 Ha: µ  100 Two-tailed Test  is divided equally between the two tails of the critical region Means less than or greater than

20. H0: µ = 100 Ha: µ  100 Two-tailed Test  is divided equally between the two tails of the critical region Means less than or greater than Reject H0 Fail to reject H0 Reject H0 100 Values that differ significantly from 100

21. H0: µ  100 Ha: µ > 100 Right-tailed Test

22. H0: µ  100 Ha: µ > 100 Right-tailed Test Points Right

23. H0: µ  100 Ha: µ > 100 Fail to reject H0 Reject H0 Right-tailed Test Points Right Values that differ significantly from 100 100

24. H0: µ  100 Ha: µ < 100 Left-tailed Test

25. H0: µ  100 Ha: µ < 100 Left-tailed Test Points Left

26. H0: µ  100 Ha: µ < 100 Left-tailed Test Points Left Reject H0 Fail to reject H0 Values that differ significantly from 100 100

27. always test the null hypothesis 1. Reject the H0 2. Fail to reject the H0 need to formulate correct wording of finalconclusion Conclusions in Hypothesis Testing

28. Wording of Final Conclusion Start Does the original claim contain the condition of equality (This is the only case in which the original claim is rejected). “There is sufficient evidence to warrant rejection of the claim that. . . (original claim).” Yes (Reject H0) Yes (Original claim contains equality and becomes H0) Do you reject H0?. No (Fail to reject H0) “There is not sufficient evidence to warrant rejection of the claim that. . . (original claim).” No (Original claim does not contain equality and becomes Ha) (This is the only case in which the original claim is supported). Yes (Reject H0) “The sample data supports the claim that . . . (original claim).” Do you reject H0? No (Fail to reject H0) “There is not sufficient evidence to support the claim that. . . (original claim).”

29. some texts use “accept the null hypothesis we are not proving the null hypothesis sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect) Accept versus Fail to Reject

30. Power of a Hypothesis Test is the probability (1 - ) of rejecting a false null hypothesis, which is computed by using a particular significance level  and a particular value of the mean that is an alternative to the value assumed true in the null hypothesis. Definition

31. for testing claims about population means 1) The sample is a simple random       sample. 2) The sample is large (n > 30). a) Central limit theorem applies b) Can use normal distribution 3) If  is unknown, we can use sample standard deviation s as estimate for . Assumptions

32. Goal Identify a sample result that is significantly different from the claimed value Traditional (or Classical) Method of Testing Hypotheses

33. The traditional (or classical) method of hypothesis testing convertsthe relevant sample statistic into a test statistic which we compare to the critical value.

34. 1.State the hypotheses 2. Decide on a model. 3. Determine the endpoints of the rejection region and state the decision rule. 4. Compute the test statistic 5. State the conclusion Hypotheses Testing 5 Step Process

35. x - µx z=  n Test Statistic for Claims about µ when n > 30 Test Statistic for Claims about µ when n < 30 x - µx t=  n

36. Reject the null hypothesis if the test statistic is in the critical region Fail to reject the null hypothesis if the test statistic is not in the critical region Decision Criterion

37. FIGURE 7-4 Wording of Final Conclusion Does the original claim contain the condition of equality (This is the only case in which the original claim is rejected). “There is sufficient evidence to warrant rejection of the claim that. . . (original claim).” Yes (Reject H0) Yes (Original claim contains equality and becomes H0) Do you reject H0?. No (Fail to reject H0) “There is not sufficient evidence to warrant rejection of the claim that. . . (original claim).” No (Original claim does not contain equality and becomes Ha) (This is the only case in which the original claim is supported). Yes (Reject H0) “The sample data supports the claim that . . . (original claim).” Do you reject H0? No (Fail to reject H0) “There is not sufficient evidence to support the claim that. . . (original claim).”

38. Example:Given a data set of 106 healthy body temperatures, where the mean was 98.2o and s = 0.62o , at the 0.05 significance level, test the claim that the mean body temperature of all healthy adults is equal to 98.6o.

39. H0 :  = 98.6o Ha :  98.6o Example:Given a data set of 106 healthy body temperatures, where the mean was 98.2o and s = 0.62o , at the 0.05 significance level, test the claim that the mean body temperature of all healthy adults is equal to 98.6o. Steps: 1) State the hypotheses 2) Determine the model Two tail Z test, n > 30

40. 3) Determine the Rejection Region = 0.05 /2= 0.025 (two tailed test) 0.4750 0.4750 0.025 0.025 z = - 1.96 1.96

41. 4) Compute the test statistic x - µ 98.2 - 98.6 z=== - 6.64  0.62 n 106

42. 5) State the Conclusion Sample data: x = 98.2o or z = - 6.64 Reject H0: µ = 98.6 Reject H0: µ = 98.6 Fail to Reject H0: µ = 98.6 z = - 1.96 µ = 98.6 or z = 0 z = 1.96 z = - 6.64 There is sufficient evidence to warrant rejection of claim that the mean body temperatures of healthy adults is equal to 98.6o. REJECT H0

43. for testing claims about population means 1) The sample is a simple random sample. 2) The sample is small (n  30). 3) The value of the population standard  deviation  is unknown. 4) The sample values come from a population  with a distribution that is approximately  normal. Assumptions

44. Critical Values Found in Table A-3 Degrees of freedom (df) = n -1 Critical t values to the left of the mean are negative Test Statistic for a Student t-distribution x -µx t = s n

45. 1. The Student t distribution is different for different sample sizes (see Figure 6-5 in Section 6-3). 2. The Student t distribution has the same general bell shape as the normal distribution; its wider shape reflects the greater variability that is expected with small samples. 3. The Student tdistribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). 4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a = 1). 5. As the sample size n gets larger, the Student t distribution get closer to the normal distribution. For values of n > 30, the differences are so small that we can use the critical z values instead of developing a much larger table of critical t values. (The values in the bottom row of Table A-3 are equal to the corresponding critical z values from the normal distributions.) Important Properties of the Student t Distribution

46. Figure 7-11 Choosing between the Normal and Student t-Distributions when Testing a Claim about a Population Mean µ Start Use normal distribution with x - µx Is n > 30 ? Yes Z / n (If  is unknown use s instead.) No Is the distribution of the population essentially normal ? (Use a histogram.) No Use nonparametric methods, which don’t require a normal distribution. Yes Use normal distribution with Is  known ? x - µx Z / n No (This case is rare.) Use the Student t distribution with x - µx t s/ n

47. The larger Student t critical value shows that with a small sample, the sample evidence must be more extreme before we consider the difference is significant.

48. 1. HO:µ = 5500 HA: µ  5500 -2.571 2.571 A company manufacturing rockets claims to use an average of 5500 lbs of rocket fuel for the first 15 seconds of operation. A sample of 6 engines are fired and the mean fuel consumption is 5690 lbs with a sample standard deviation of 250 lbs. Is the claim justified at the 5% level of significance? 2. Two tail t test, n < 30, unknown population standard deviation 1.862 3. t critical for 5% for a two tail test with 5 d.f. is 2.571 • Fail to reject HO, there is no evidence at the .05 level • that the average fuel consumption is different from µ = 5500 lbs

49. Table A-3 includes only selected values of  Specific P-values usually cannot be found Use Table to identify limits that contain the P-value Some calculators and computer programs will find exact P-values P-Value Method

50. very similar to traditional method key difference is the way in which we decide to reject the null hypothesis approach finds the probability (P-value) of getting a result and rejects the null hypothesis if that probability is very low P-Value Methodof Testing Hypotheses