Jury TrialTruthTest of hypothesisTruthScientifically significant vs. Statistically significantHYPOTHESIS TESTINGExample: Average intake in children for dietary fat is 70 g of fat per day. Suppose we want to study children who eat a vegetarian diet. Possible hypotheses are1. Average fat intake is 70 g per day2. Average fat intake is less than 70 g per dayDef: One-sample problem—a single distribution.Def: Hypothesis—statement about parameters in a population or populations. We want to know how likely this is to be true, given the evidence (data). For example,1. Average number of beds filled per day in the hospital2. Average number of minutes per day the doctor spends with a patient3. Average lead content of water for a housing projectDef: Null hypothesis—Ho—the hypothesis to be tested. This is usually a statement of no difference. The population value of the parameter is not different from some specified value. Def: Alternative hypothesis—H1 or HA—This is the statement we will accept if we reject the null hypothesis. Ho: Mean fat intake in children on a vegetarian diet is 70 g per day. Ho:  = 0 or  0H1: Mean fat intake in children on a vegetarian diet is < 70 g per day. H1:  < 0Possible decisions:1. Accept Ho (really, fail to reject Ho)2. Reject HoPossible Scenarios:Jury TrialTruthVerdict Innocent GuiltyInnocent Correct decision ErrorGuilty Error Correct decisionTest of hypothesisTruthResults of Test HoH1Accept HoCorrect decision Type II errorReject HoType 1 error Correct decisionDef: Type I error is the probability of rejecting Ho when Ho is true.Def: Type II error is the probability of accepting Ho when H1 is true.Example: We have developed a new procedure to improve survival of premature infants. If the hospital adopts these procedures, there will have to be new rooms and new equipment purchased. This is very costly.H0: There is no improvement in survival of premature infants.H1: There is improvement in survival of premature infants1. What does a Type I error imply?2. What does a Type II error imply?Def: Level of significance— = Probability of a Type I error. This is the area under the curve below (or above) the critical value. This is the probability of rejecting Ho when Ho is true. Def: —Probability of a Type II error.Def: 1- —Power of a test. This is the Pr(rejecting Ho|H1 is true). Goal: Make ,  as small as possible. Usually, as ,  and as , . Fix  (0.05 or 0.01). Find a test to minimize .Best test for the fat experiment is one based on X.Def: Acceptance Region—These are the values of X for which Ho is accepted.Def: Rejection Region—These are the values of X for which Ho is rejected. Note: For this example, we are conducting a one-sided or one-tailed test. We will only reject Ho for values of X that are low. Def: One-tailed—This is a test in which values of parameter under H1 either > or < values under Ho but not both. How small should X be to reject Ho? Set the significance level to . Reject Ho for all values of X < c. Select c so that Type I error is . We define “c” in terms of standardized values, i.e., subtract 0 and divide by ns/.10n~tns/μXtIf t < tn-1,  reject Ho.If t  tn-1,  accept Ho. This is the t-test for the mean of a normal distribution with unknown population variance. We estimate the variance with the sample variance. Def: t is an example of a test statistic. Many test statistics are of the formobservedoferrorstandardedhypothesizobserved Def: tn-1,  is called the critical value. Example: We want to test whether the mean intake of vegetarian children is less than 70 g per day. Conduct the test at the level of significance of 0.05 (5%). Suppose we sample 20 children who are vegetarians and found X = 65 g per day with s = 10 g per day. 707010::HHFrom Table 5, t19, 0.05 = -1.72924223625201070650../ns/μXtDecision: Reject HoConclusion: We have evidence to reject the null hypothesis. It appears that vegetarian children have dietary fat intake that is less than 70 g per day. Note: If  = 0.01, t19, 0.01 = -2.539.What would decision and conclusion be? Def: p -value—This is the  level whereby we are indifferent between accepting or rejecting Ho, given the sample data. This is the probability of getting a value of the test statistic this extreme or more extreme, given Ho is true. p = Pr(|t|  tn-1)Note: t19, 0.025 = -2.093t19, 0.01 = -2.539t = -2.24Therefore, 0.01  p  0.025The p-value tells us how significant our results are without repeated tests at different  levels. Suggested terminology:0.01  p  0.05 Statistically significant0.001  p  0.01 Highly statistically significantp < 0.001 Very highly statistically significantp > 0.05 Not statistically significantScientifically significant vs. Statistically significantExample: Suppose HDL cholesterol levels in males 45-54 years old are normally distributed with  = 42 mg/dl. Suppose 61 men enroll in an exercise programfor 12 weeks. Suppose that after the exercise program, their average HDL cholesterol level is 45 mg/dl with s = 8 mg/dl. Test the hypothesis that the mean HDL cholesterol level is higher than in the population. 424210::HHLet  = 0.05.Critical Value: tn-1, 1- = t60, 0.95 = 1.671Reject Ho if t > 1.671.932024136184245../tDecision: Conclusion: