Psych 230 Psychological Measurement and Statistics

Psych 230Psychological 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? • State the hypotheses • Design the experiment • Collect the data • Create the statistical hypotheses • Select the appropriate statistical test • Decide the size of the rejection region (value of ) • Calculate the obtained and critical values • 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 (H0) • there is no difference in the groups we are testing • The Alternative Hypothesis (H1) • there is a real difference in the groups we are testing

Hypothesis Testing - Experiment • The Null Hypothesis (H0) • there is no difference in the groups we are testing • H0 : 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 • H1 : 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 SX = 3.03

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

Statistical Hypotheses • Our hypotheses were: • H0 : People born in March are the same height as those born in other months • H1 : People born in March are not the same height as those born in other months • H0 :  = X • H1 :  ≠ X

Statistical Hypotheses • H0 :  = X  a a a a

Statistical Hypotheses • H0 :  = X  X a a a a

Statistical Hypotheses • H1 :  ≠ X  a a a a

Statistical Hypotheses • H1 :  ≠ X  X a a a a

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 H0 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 (H1) • People born in March are not the same height as others (H1) • A one-tailed test is used when you predict the specific direction in which scores will change • Prozac will improve depression symptoms (H1)

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

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

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

The obtained and critical value  a a a 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. • Zcrit = 1.96

Zcrit  Zcrit=-1.96 Zcrit=+1.96 a a a a

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

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

Zcrit and Zobt Zobt<Zcrit  Zobt=-0.57 Zcrit=-1.96 Zcrit=+1.96 a a a a

Drawing a conclusion • Zobt<Zcrit therefore we do not reject H0 • We do not have enough evidence to say that our null hypothesis is false • When we fail to reject H0 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 H0, we do not prove that H0 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 Zobt = -2.03

Zcrit and Zobt  Zobt=-2.03 Zcrit=-1.96 Zcrit=+1.96 a a a a

Drawing a different conclusion • Zobt>Zcrit therefore we reject H0 • When we reject H0 and accept H1 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 (zobt) 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 H0 when H0 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 a

Type 2 Error • A Type II error is defined as retaining H0 when H0 is false (and H1 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 b 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). • Is this a one-tailed or two-tailed test? Why? • What are H0 and H1? • Compute zobt • With =0.05, what is zcrit? • 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). • 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). • 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.

Psych 230 Psychological Measurement and Statistics