Chapter 8: Introduction to Hypothesis Testing

2 Hypothesis Testing An inferential procedure that uses sample data to evaluate the credibility of a hypothesis about a population.

3 Before and after treatment comparisons µ = 18 µ = ?  = 4 Known population before treatment Unknown population after treatment TreatmentTreatment

4 Hypothesis Testing Step 1: State hypothesis Step 2: Set criteria for decision Step 3: Collect sample data Step 4: Evaluate null hypothesis Step 5: Conclusion

5 Hypothesis testing example problem µ = 18 g  = 4 g Step 1: Ho:Ho:µ = 18 g Weight of rats of alcoholic mothers (There is no effect of alcohol on the average birth weight) H1:H1: µ  18 g (There is an effect…)  = 0.05 Step 2: Set criteria z Critical Region z > 1.96 z < or Step 3:n = 25 ratsX = 15.5 g Step 4: Reject H o because Z obt of is in the critical region. Step 5: Conclusion

6 Error Type / Correct Decision table Type I Error Type II Error Decision Correct Reject H o Retain H o Experimenter’s Decision Actual Situation No Effect, H o True Effect Exists, H o False

7 Jury’s Verdict vs. Actual Situation Type I Error Type II Error Verdict Correct Guilty Innocent Jury’s Verdict Actual Situation Did Not Commit Crime Committed Crime

8 Fixed coin (cheating) vs. OK coin (fair) Type I Error Type II Error Correct Decision Coin Fixed (Cheating) Coin O.K. (Fair) Your Decision Actual Situation Coin O.K. (Fair) Coin Fixed (Cheating)

9 The distribution of sample means (all possible experimental outcomes) if the null hypothesis is true Reject H o Extreme values (probability < alpha) if H o is true Possible, but “very unlikely”, outcomes

10 Evaluating Hypotheses Reject H o Middle 95% Very Unlikely X µ z

11 Infant handling problem We know from national health statistics that the average weight for 2-year olds is µ = 26 lbs. and  = 6. A researcher is interested in the effect of early handling on body weight/growth. A sample of 16 infants is taken. Each parent is instructed in how to provide additional handling. When the infants are 2-years old, each child is weighed,

12 Infant problem: Before and after treatment comparisons µ = 26 µ = ?  = 6 Known population before treatment Unknown population after treatment TreatmentTreatment

13 Hypothesis testing: Infant handling problem µ = 26 lbs.  = 4 lbs. Step 1: Ho:Ho:µ = 26 lbs. Handled infants (There is no effect of extra handling on average weight of 2 year olds) H1:H1: µ  26 lbs. (There is an effect of…)  = 0.01 Step 2: Set criteria z Critical Region z > 2.58 z < or Step 3:n = 16 infantsX = 31 pounds Step 4: Reject H o because Z obt of 3.33 is in the critical region. Step 5: Handling infants more during infancy significantly changes their weight, z = 3.3, p < 0.01.

14 Evaluating Infant Handling Hypothesis Reject H o Middle 99% Very Unlikely X µ z

15 Hypothesis testing procedure 1.State hypothesis and set alpha level 2.Locate critical region e.g.z > | 1.96 | z > 1.96 or z < Obtain sample data and compute test statistic 4. Make a decision about the H o 5. State the conclusion

16 Reject H o Null Hypothesis Rejection areas on distribution curve p >  p < 

17 Normal curve with different alpha levels z  =.05  =.01  =.001

18 Assumptions for Hypothesis Tests with z-scores: 1.Random sampling 2.Value of  unchanged by treatment 3.Sampling distribution normal 4.Independent observations

19 Normal curve with null rejection area on negative side µ = 10 Reject H o Very unlikely if H o is true Expect X around 10 or larger if H o is true

20 Reading Improvement Problem A researcher wants to assess the “miraculous” claims of improvement made in a TV ad about a phonetic reading instruction program or package. We know that scores on a standardized reading test for 9-year olds form a normal distribution with µ = 45 and  = 10. A random sample of n = 25 8-year olds is given the reading program package for a year. At age 9, the sample is given the standardized reading test.

21 Normal curve with rejection area on the positive side Reject H o Data indicates that H o is wrong 45 µ z X

22 Two-tailed vs. One-tailed Tests 1.In general, use two-tailed test 2.Journals generally require two-tailed tests 3.Testing H o not H 1 4.Downside of one-tailed tests: what if you get a large effect in the unpredicted direction? Must retain the H o

23 Error and Power Type I error =  –Probability of a false alarm Type II error =  –Probability of missing an effect when H o is really false Power = 1 -  –Probability of correctly detecting effect when H o is really false

24 Factors Influencing Power (1 -  ) 1.Sample size 2.One-tailed versus two-tailed test 3.Criterion (  level) 4.Size of treatment effect 5.Design of study

25 Distributions demonstrating power at the.05 and.01 levels  Treatment Distribution Null Distribution  =.05 Reject H o X  Treatment Distribution Null Distribution  =.01 Reject H o X 1 -  Reject H o Reject H o 1 -  µ = 180 µ = 200 (a) (b)

26 Distributions demonstrating power with different size n’s  Treatment Distribution Null Distribution n = 25 Reject H o X 1 -  Reject H o µ = 180µ = 200 (a)  Treatment Distribution Null Distribution n = 100 Reject H o X 1 -  (b) Reject H o

27 Large distribution demonstrating power  Treatment Distribution Null Distribution Reject H o X 1 -  Reject H o µ 180 µ 200

28 Large distribution demonstrating power (with closer means)  Treatment Distribution Null Distribution Reject H o X 1 -  Reject H o µ 190 µ 200

29 Frequency distributions Frequency Frequency X X

30 Are birth weights for babies of mothers who smoked during pregnancy significantly different? µ = 2.9 kg  = 2.9 kg Random Sample:n = , 2.0, 2.2, 2.8, 3.2, 2.2, 2.5, 2.4, 2.4, 2.1, 2.3, 2.6, 2.0, 2.3

31 The distribution of sample means if the null hypothesis is true (all the possible outcomes) Sample means close to H o : high-probability values if H o is true µ from H o Extreme, low-probability values if H o is true Extreme, low-probability values if H o is true

32 Distribution of sample means based on percentages Middle 95%: high-probability values if H o is true µ from H o Reject H o z = z = 1.96 Critical Region: Extreme 5%

33 One-tailed distribution of means 26 µ z X Reject H o Data indicates that H o is wrong

34 Population comparison before and after alcohol µ = 18 Normal  = 4 µ = ? Normal  = 4 AlcoholAlcohol Sample n = 16 X = 15 Population with alcohol Population without alcohol

35 Population according to the null hypothesis Normal µ = 18  = 4 µ = 18 Decision criteria for the hypothesis test Reject H o Middle 95% High probability values if H o is true Fail to reject H o z = -1.96