Design and Data Analysis in Psychology I Salvador Chacón Moscoso Susana Sanduvete Chaves School of Psychology Dpt. Experimental Psychology 1

Lesson 8 Null hypothesis significance testing 2 Pérez, J., Manzano, V., & Fazeli, H. (1999). Análisis de datos en Psicología. Madrid: Pirámide.

 Statistical decision: procedure to choose an alternative based on statistics. E.g., is cognitive therapy an useful treatment to reduce anxiety? 1. Introduction 3 GroupExperimentalControl Mean810

2. Steps 4 1. Concrete the null hypothesis (H 0 ): there are not differences between groups; there is not relationship between variables. 2. Calculations to check if H 0 is true or false. When H 0 is rejected, then the alternative hypothesis (H 1 ) is accepted: there are differences between groups; there is relationship between variables. 3. Take a final decision, assuming some level of risk (probability of being wrong in your decision).

 Similarity: both are based on the same probabilistic theory or background  Differences: 3. Estimation vs. decision 5 EstimationDecision Problems that solves Calculate the percentage of 14-year-old smokers Are there more than 60% of 14- year-old smokers? SolutionA valueYes/no

4. Statistical decision. Example 6 One year ago, the opinion about abortion in the population presented a mean of Nowadays, we obtained a mean of 12 from a sample of 200 participants with a standard deviation of 7.2. Has the opinion changed? (α, the probability of being wrong when you accept the H 0,is 0.05).

4. Statistical decision. Example 7 Steps: 1. Concrete the null hypothesis (H 0 ). 2. Calculations to check if H 0 is true or false: 3. Take a final decision: - If the mean is in the interval = H 0 is accepted (there are not differences). - If the mean is not in the interval = H 0 is rejected (there are differences).

4. Statistical decision. Example 8 Steps: 1. H 0 : there are not differences; the opinion has not changed. 2. Calculations to check if H 0 is true or false (two ways):

4. Statistical decision. Example 9 3. Take a final decision (two ways): The mean (12) is not in the interval [ ] or The mean (10.57) is not in the interval [11-13] - H 0 is rejected. There are differences. The opinion about abortion has statistically changed in one year H0H0 H1H1 H1H1

5. Statistical significance 10  It is the probability of obtaining values out of the interval calculated (the most extreme values).  When p ≤ α → H 0 is rejected.  When p > α → H 0 is accepted.

5. Statistical significance 11  Decisions about the H 0 can be taken based on the confidence interval or the statistical significance.  In both cases, the decision is going to be always the same,  So you do not have to do both procedures; one is enough.

5. Statistical significance. Example 1 12  In the previous example, calculate the statistical significance

5. Statistical significance. Example

5. Statistical significance. Example 1 14  The probability of obtaining a value out of the interval is  p ≤ α < 0.05 → H 0 is rejected. There are differences. The opinion about abortion has statistically changed in one year.  When we used the confidence interval to take the decision about the H 0, the conclusion obtained was the same.

5. Statistical significance. Example 2 15  Previous studies found that the average intellectual coefficient in children between 10 and 12 years old is 105. In a sample of 30 children, the mean obtained was 113 and its standard deviation, Can we consider that children between 10 and 12 years old have an intellectual coefficient of 105? (α = 0.05).

5. Statistical significance. Example

5. Statistical significance. Example 2 17  p ≤ α < 0.05 → H 0 is rejected. There are differences. We can not consider that children between 10 and 12 years old have an intellectual coefficient of 105.

5. Statistical significance. Example 3 18  The 73% of the population presented at least one anxiety attack. 70 adults participated in a relaxation program. 30 of them never suffered an anxiety attack. Is the program effective? Does the program decrease significantly the number of anxiety attacks? (α = 0.05).

5. Statistical significance. Example If 30 did not suffer an anxiety attack, then 40 suffered it <0.73, so the program could be effective.

5. Statistical significance. Example 3 20  The probability of obtaining a value out of the interval is  p ≤ α < 0.05 → H 0 is rejected. There are differences. We can consider that the program is effective. It decreases significantly the number of anxiety attacks.

6. Decisions about α  The value of α: Depends on the consequences of being wrong in your decision when you reject the null hypothesis. Higher seriousness implies lower α; e.g. Is a treatment against cancer effective? Vs. Is a program to learn to read effective? Should be chosen before the statistical analysis to increase the credibility of the results and decisions taken based on them. 21

7. Standardized distance  The same decision can be taken comparing Z scores instead of statistical significances (so you do not need to use the tables).  When Z 0 ≤ Z α/2 → H 0 is accepted. There are not differences between groups. There is not relationship between variables.  When Z 0 > Z α/2 → H 0 is rejected. There are differences between groups. There is relationship between variables. 22

23 0 Z α/2 H0H0 H1H1 7. Standardized distance

7. Standardized distance. Example 1  Take the decision in the previous exercise 1, concluding with Z scores (problem in slide 6; Z 0 in slide 13). 24

7. Standardized distance. Example 1  Z 0 = 2.8  2.8 > 1.96 → H 0 is rejected. There are differences. The opinion about abortion has statistically changed in one year Z α/2 =1.96 Z 0 =2.8

7. Standardized distance. Example 2  Take the decision in the previous exercise 2, concluding with Z scores (problem in slide 15; Z 0 in slide 16). 26

7. Standardized distance. Example 2  Z 0 = 2.1  2.8 > 1.96 → H 0 is rejected. There are differences. We can not consider that children between 10 and 12 years old have an intellectual coefficient of Z α/2 =1.96 Z 0 =2.1

7. Standardized distance. Example 3  Take the decision in the previous exercise 3, concluding with Z scores (problem in slide 18; Z 0 in slide 19). 28

7. Standardized distance. Example 3  Z 0 =  3.01 > 1.96 → H 0 is rejected. There are differences. We can consider that the program is effective. It decreases significantly the number of anxiety attacks Z α/2 =1.96 Z 0 =3.01 Z α/2 =-1.96 Z 0 =-3.01

7. Standardized distance. Example 4  Spanish books about Psychology have a mean of 250 pages. Taking into account that I have 42 books with a mean of 280 pages and a standard deviation of 100 pages, can I conclude that my books are larger than usual? (α = 0.05). 30

7. Standardized distance. Example Z 0 =1.92 Z α/2 =1.96

7. Standardized distance. Example 4 32  1.92 < 1.96 → H 0 is accepted. There are not differences. I can not conclude that my books are statistically larger than usual.