Psych 230 Psychological Measurement and Statistics Pedro Wolf October 21, 2009.

Psych 230 Psychological Measurement and Statistics Pedro Wolf October 21, 2009

Today…. Hypothesis testing Null and Alternative Hypotheses Z-test Significant and Nonsignificant results Types of Statistical error (type 1 and type 2)

A scientific question A biology professor studies the effect of nutrition on physical attributes. He theorizes that maternal nutrition can affect the height and weight of their offspring. Further, he thinks that the time of year a child is conceived, due to seasonal nutrition factors, has a relationship with how tall that child will be. After study, he establishes that yearly nutrition is most different from the norm in June. So, he wants to know whether people conceived in June are a different height than the population.

How do we answer this question? 1. State the hypotheses 2. Design the experiment 3. Collect the data 4. Create the statistical hypotheses 5. Select the appropriate statistical test 6. Decide the size of the rejection region (value of  ) 7. Calculate the obtained and critical values 8. Make our conclusion

Hypothesis Testing Experimental hypotheses describe the predicted outcome we may or may not find in an experiment As scientists, we try to be conservative –we should assume no effect of what we are observing or testing Does Prozac work in treating depression? Are men better at math than women? Do we learn better when practice is all at once or spread over time?

Hypothesis Testing In experiments, we usually identify two hypotheses The Null Hypothesis (H 0 ) –there is no difference in the groups we are testing The Alternative Hypothesis (H 1 ) –there is a real difference in the groups we are testing

Hypothesis Testing - Experiment The Null Hypothesis (H 0 ) –there is no difference in the groups we are testing H 0 : People born in March are the same height those born in other months The Alternative Hypothesis –there is a real difference in the groups we are testing H 1 : People born in March are not the same height as those born in other months

Designing the Experiment/Study Dependent/Observed Variable? We want to measure height Independent/Predictor Variable? Month of birth March vs all other months Observational or Experimental study?

Height - Sample Heights of those born in March: 63, 64, 62, 67, 68, 66, 72, 64 Calculate the mean and standard deviation: X = 65.75 S X = 3.03

Our Data Population  = 66.57  x = 4.091 Sample N = 8 X = 65.75 S X = 3.03

Statistical Hypotheses Our hypotheses were: –H 0 : People born in March are the same height as those born in other months –H 1 : People born in March are not the same height as those born in other months H 0 :  = X H 1 :  ≠ X

Statistical Hypotheses aaa  a H 0 :  = X

Statistical Hypotheses aaa  a H 0 :  = X X

Statistical Hypotheses aaa  a H 1 :  ≠ X

Statistical Hypotheses aaa  a H 1 :  ≠ X X

Z-Test The z-test is the procedure for computing a z- score for a sample mean on the sampling distribution of means Comparing a sample to a population

Assumptions of the Z-Test We have randomly selected one sample The dependent variable is at least approximately normally distributed in the population and involves an interval or ratio scale We know the mean of the population of raw scores We know the true standard deviation of the population

Deciding the size of the rejection region Usually, Psychologists use a rejection region of 0.05 –known as  (alpha) –sometimes use 0.01 or 0.001 –If the H 0 is true, the probability of getting an xbar this extreme is 

One-tail versus two-tail testing A two-tailed test is used when we predict that there is a relationship, but we do not specifically predict the direction in which scores will change –Freshmen and Seniors score differently in tests of sociability (H 1 ) –People born in March are not the same height as others (H 1 ) A one-tailed test is used when you predict the specific direction in which scores will change –Prozac will improve depression symptoms (H 1 )

Rejection region When a two-tailed test is used, we need to spread our  value across both tails of the distribution When a one-tailed test is used, all our  value is put in one tail of the distribution

Rejection region aaa  a A criterion of 0.05 and a region of rejection in two tails

Rejection region aaa  a A criterion of 0.05 and a region of rejection in just one tail

Rejection region - Experiment In our study, we will use  =0.05 This is a two-tailed test Therefore we will have  =0.025 in each tail

Rejection region aaa  a A criterion of 0.05 and a region of rejection in two tails (0.025 in each tail)

The obtained and critical value aaa  a

Calculate the critical value We want the z-score that corresponds to an area in the tail of 0.025 Look up tables, Starts page 548: Where area beyond z=0.025. Z crit =  1.96

Z crit aaa  a Z crit =-1.96Z crit =+1.96

Calculate the obtained value Population  = 66.57  x = 4.09 Sample N = 8 X = 65.75 S X = 3.03

Calculate the obtained value  x = 4.09 / √8 = 1.44 Z obt = (65.75 - 66.57) / 1.44 = -0.57

Z crit and Z obt aaa  a Z obt < Z crit Z crit =-1.96Z crit =+1.96 Z obt =-0.57

Drawing a conclusion Z obt < Z crit therefore we do not reject H 0 We do not have enough evidence to say that our null hypothesis is false When we fail to reject H 0 we say the results are nonsignificant. Nonsignificant indicates that the results are likely to occur if the predicted relationship does not exist in the population We conclude that people born in March are no different in height from those born in other months

Drawing a conclusion When we fail to reject H 0, we do not prove that H 0 is true Nonsignificant results provide no convincing evidence - one way or the other - as to whether a relationship exists in nature

Drawing a different conclusion Let’s assume Z obt = -2.03

Z crit and Z obt aaa  a Z crit =-1.96Z crit =+1.96 Z obt =-2.03

Drawing a different conclusion Z obt > Z crit therefore we reject H 0 When we reject H 0 and accept H 1 we say the results are significant. Significant indicates that the results are too unlikely to occur if the predicted relationship does not exist in the population We conclude that people born in March are significantly shorter in height than those born in other months

Z scores and p values Z scores can be readily changed back into proportions, and probabilities When reporting the results of tests, a z value (z obt ) and p value are often reported In future homework assignments you’ll need to use p values in the homework

Types of Statistical Error When conducting a statistical test, we can make two kinds of errors: Type 1 Type 2

Type 1 Error A Type I error is defined as rejecting H 0 when H 0 is actually true In a Type I error, we conclude that the predicted relationship exists when it really does not The probability of a Type I error equals 

Type 2 Error A Type II error is defined as retaining H 0 when H 0 is false (and H 1 is actually true) In a Type II error, we conclude that the predicted relationship does not exist when it really does The probability of a Type II error is  or 1-p

Problem 1 Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 1. Is this a one-tailed or two-tailed test? Why? 2. What are H 0 and H 1 ? 3. Compute z obt 4. With  =0.05, what is z crit ? 5. What conclusion should we draw from this study?

Problem 1 Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 1. Is this a one-tailed or two-tailed test? Why?

Problem 1 Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 1. Is this a one-tailed or two-tailed test? Why? It will be a two-tailed test, as we are not predicting the direction that the scores will change. That is, we are asking whether music leads to a different performance, either better or worse.

Problem 1 Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 2.What are H 0 and H 1 ?

Problem 1 Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 2.What are H 0 and H 1 ? H 0 : Music does not lead to changes in test performance H 0 :  music = 50 H 1 : Music leads to changes in test performance H 1 :  music  50

Problem 1 Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 3.Compute z obt

Problem 1 Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 3. Compute z obt  x = 12 / √49 = 1.714 Z obt = (54.63 - 50) / 1.714 = +2.70

Problem 1 Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 4. With  =0.05, what is z crit ?

Problem 1 Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 4. With  =0.05, what is z crit ? A two-tailed test with  =0.05, leaves 0.025 in each tail. The Z-score for 0.025 is 1.96. Therefore z crit =  1.96

Z crit and Z obt aaa  a Z crit =-1.96Z crit =+1.96 Z obt =+2.70

Problem 1 Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 5. What conclusion should we draw from this study?

Problem 1 Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 5. What conclusion should we draw from this study? As z obs > z crit, we reject H 0 and tentatively accept H 1. People listening to music while taking a test score significantly better than those not listening to music.

Problem 1(b) Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 6.Based on our conclusion (music affects test scores), what is the probability we made a Type 1 error? What would that error be? What is the probability we made a Type 2 error?

Problem 1(b) Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 6.Based on our conclusion (music affects test scores), what is the probability we made a Type 1 error? What would that error be? What is the probability we made a Type 2 error? Probability of a Type 1 error is 0.05 (our  value). This error would be saying a relationship existed between music and test scores when really it did not.

Problem 1(b) Listening to music while taking a test may be relaxing or distracting. To determine which, 49 participants are tested while listening to music and they produce a mean score of 54.63. In the population, the mean score without music is 50 (std dev of 12). 6.Based on our conclusion (music affects test scores), what is the probability we made a Type 1 error? What would that error be? What is the probability we made a Type 2 error? As we rejected H 0, there is no probability of a Type 2 error This would be saying there is no relationship when there really was.

Problem 2 A researcher asks whether attending a private school leads to higher or lower performance on a test of social skills. A sample of 100 students from a private school produces a mean score of 71.30. The  for students from public schools is 75.62 (  x =28). 1. Is this a one-tailed or two-tailed test? Why? 2. What are H 0 and H 1 ? 3. Compute z obt 4. With  =0.01, what is z crit ? 5. What conclusion should we draw from this study?

Problem 2 A researcher asks whether attending a private school leads to higher or lower performance on a test of social skills. A sample of 100 students from a private school produces a mean score of 71.30. The  for students from public schools is 75.62 (  x =28). 1. Is this a one-tailed or two-tailed test? Why?

Problem 2 A researcher asks whether attending a private school leads to higher or lower performance on a test of social skills. A sample of 100 students from a private school produces a mean score of 71.30. The  for students from public schools is 75.62 (  x =28). 1. Is this a one-tailed or two-tailed test? Why? It will be a two-tailed test, as the researcher is looking for higher or lower scores.

Problem 2 A researcher asks whether attending a private school leads to higher or lower performance on a test of social skills. A sample of 100 students from a private school produces a mean score of 71.30. The  for students from public schools is 75.62 (  x =28). 2.What are H 0 and H 1 ?

Problem 2 A researcher asks whether attending a private school leads to higher or lower performance on a test of social skills. A sample of 100 students from a private school produces a mean score of 71.30. The  for students from public schools is 75.62 (  x =28). 2.What are H 0 and H 1 ? H 0 : Public and private students score the same on tests of social skills H 0 :  private = 75.62 H 1 : Public and private students score differently on tests of social skills H 1 :  private  75.62

Problem 2 A researcher asks whether attending a private school leads to higher or lower performance on a test of social skills. A sample of 100 students from a private school produces a mean score of 71.30. The  for students from public schools is 75.62 (  x =28). 3.Compute z obt

Problem 2 A researcher asks whether attending a private school leads to higher or lower performance on a test of social skills. A sample of 100 students from a private school produces a mean score of 71.30. The  for students from public schools is 75.62 (  x =28). 3. Compute z obt  x = 28 / √100 = 2.8 Z obt = (71.30 - 75.62) / 2.8 = -1.54

Problem 2 A researcher asks whether attending a private school leads to higher or lower performance on a test of social skills. A sample of 100 students from a private school produces a mean score of 71.30. The  for students from public schools is 75.62 (  x =28). 4. With  =0.01, what is z crit ?

Problem 2 A researcher asks whether attending a private school leads to higher or lower performance on a test of social skills. A sample of 100 students from a private school produces a mean score of 71.30. The  for students from public schools is 75.62 (  x =28). 4. With  =0.01, what is z crit ? A two-tailed test with  =0.01, leaves 0.005 in each tail. The Z-score for 0.005 is 2.57. Therefore z crit =  2.57

Z crit and Z obt aaa  a Z crit =-2.57Z crit =+2.57 Z obt =-1.54

Problem 2 A researcher asks whether attending a private school leads to higher or lower performance on a test of social skills. A sample of 100 students from a private school produces a mean score of 71.30. The  for students from public schools is 75.62 (  x =28). 5. What conclusion should we draw from this study?

Problem 2 A researcher asks whether attending a private school leads to higher or lower performance on a test of social skills. A sample of 100 students from a private school produces a mean score of 71.30. The  for students from public schools is 75.62 (  x =28). 5. What conclusion should we draw from this study? As z obs < z crit, we retain H 0 The researcher has no evidence of a relationship between the type of school attended (public or private) and social skills.

Problem 3 A researcher measures the self-esteem scores of a sample of 9 statistics students, reasoning that their frustration with this course may lower their self-esteem relative to the typical college student where  =55 and  x =11.35. This sample has a mean self-esteem score of 35.11. 1. Is this a one-tailed or two-tailed test? Why? 2. What are H 0 and H 1 ? 3. Compute z obt 4. With  =0.05, what is z crit ? 5. What conclusion should we draw from this study?

Problem 3 A researcher measures the self-esteem scores of a sample of 9 statistics students, reasoning that their frustration with this course may lower their self-esteem relative to the typical college student where  =55 and  x =11.35. This sample has a mean self-esteem score of 35.11. 1. Is this a one-tailed or two-tailed test? Why?

Problem 3 A researcher measures the self-esteem scores of a sample of 9 statistics students, reasoning that their frustration with this course may lower their self-esteem relative to the typical college student where  =55 and  x =11.35. This sample has a mean self-esteem score of 35.11. 1. Is this a one-tailed or two-tailed test? Why? It will be a one-tailed test, as the researcher is just interested in lower self-esteem scores.

Problem 3 A researcher measures the self-esteem scores of a sample of 9 statistics students, reasoning that their frustration with this course may lower their self-esteem relative to the typical college student where  =55 and  x =11.35. This sample has a mean self-esteem score of 35.11. 2.What are H 0 and H 1 ?

Problem 3 A researcher measures the self-esteem scores of a sample of 9 statistics students, reasoning that their frustration with this course may lower their self-esteem relative to the typical college student where  =55 and  x =11.35. This sample has a mean self-esteem score of 35.11. 2.What are H 0 and H 1 ? H 0 : Statistics students do not differ from others in terms of their self-esteem H 0 :  stats >= 55 H 1 : Statistics students have lower self-esteem than other students H 1 :  stats < 55

Problem 3 A researcher measures the self-esteem scores of a sample of 9 statistics students, reasoning that their frustration with this course may lower their self-esteem relative to the typical college student where  =55 and  x =11.35. This sample has a mean self-esteem score of 35.11. 3.Compute z obt

Problem 3 A researcher measures the self-esteem scores of a sample of 9 statistics students, reasoning that their frustration with this course may lower their self-esteem relative to the typical college student where  =55 and  x =11.35. This sample has a mean self-esteem score of 35.11. 3. Compute z obt  x = 11.35 / √9 = 3.783 Z obt = (35.11 - 55) / 3.783 = -5.26

Problem 3 A researcher measures the self-esteem scores of a sample of 9 statistics students, reasoning that their frustration with this course may lower their self-esteem relative to the typical college student where  =55 and  x =11.35. This sample has a mean self-esteem score of 35.11. 4. With  =0.05, what is z crit ?

Problem 3 A researcher measures the self-esteem scores of a sample of 9 statistics students, reasoning that their frustration with this course may lower their self-esteem relative to the typical college student where  =55 and  x =11.35. This sample has a mean self-esteem score of 35.11. 4. With  =0.05, what is z crit ? A one-tailed test with  =0.05, looking for lower scores, leaves 0.05 in the lower tail. The Z-score for 0.05 is 1.645. Therefore z crit = -1.645

Z crit and Z obt aaa  a Z crit =- 1.645 Z obt =-5.26

Problem 3 A researcher measures the self-esteem scores of a sample of 9 statistics students, reasoning that their frustration with this course may lower their self-esteem relative to the typical college student where  =55 and  x =11.35. This sample has a mean self-esteem score of 35.11. 5. What conclusion should we draw from this study?

Problem 3 A researcher measures the self-esteem scores of a sample of 9 statistics students, reasoning that their frustration with this course may lower their self-esteem relative to the typical college student where  =55 and  x =11.35. This sample has a mean self-esteem score of 35.11. 5. What conclusion should we draw from this study? As z obs < z crit, we reject H 0 The researcher has evidence of a relationship. Statistics students score significantly lower than other college students on tests of self-esteem.

Psych 230 Psychological Measurement and Statistics Pedro Wolf October 21, 2009.

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