Psych 230 Psychological Measurement and Statistics

Psych 230Psychological Measurement and Statistics Pedro Wolf October 28, 2009

Last Time…. • Hypothesis testing • Statistical Errors • Z-test

This Time…. • T-Test • Confidence intervals • Practice problems

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 assume no relationship • The Null Hypothesis (H0) • there is no relationship between the variables • The Alternative Hypothesis (H1) • there is a real relationship between the variables

Steps to Hypothesis testing • 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

Statistical tests (so far) • The statistical tests we have used so are concentrate on finding whether a sample is representative of a known population • Two characteristics of these tests: • one sample is drawn • we know the population mean • Z-test • we also know the population variance • T-test (one sample) • we do not know the population variance

Statistical Testing • Decide which test to use • State the hypotheses (H0 and H1) • Calculate the obtained value • Calculate the critical value (size of ) • Make our conclusion

One sample T-test

One-sample T-test • We use the one sample T-test when we do not know the population variance • Only differences from before: • Tobt uses a slightly different formula • Tcrit comes from a different distribution (the T-distribution), and so we need different tables to get this value

The T-test - summary • Create H0 and H1 • Compute tobt • Compute X and s2x • Compute sx • Compute tobt • Find tcrit by using the T-tables with df = N - 1 • Compare tobt to tcrit

The T-value of our sample (Tobt) • Calculating Tobt Tobt = X - µ sx sx = √(s2x / N) : estimated standard error of the mean In General: Test statistic= Observed - Expected Standard Error

The T-distribution (Tcrit) • When using z-scores, we always looked at the same distribution (the Z-distribution) • The T-distribution is actually a family of curves, all which look slightly different depending on how many samples were used to create them • Therefore, as N changes, the exact curve we will use will change • For small samples (a small N) the curve is only roughly similar to the standard normal curve • Large samples (a big N) look very close to the standard normal curve

The T-distribution (Tcrit) • Two different T-distributions

The T-distribution (Tcrit) • We choose the curve to sample from based on not N exactly, but rather the quantity N-1 • This is termed the degrees of freedom (df) • Degrees of freedom: the number of observations in a set of data that are variable • The larger the df, the closer the t-distribution resembles a standard normal curve • When df > 120, the t-distribution is virtually identical to the standard curve, and in fact tcrit = zcrit

The T-distribution (Tcrit) • To decide whether the observed value (Tobs) is in the region of rejection, we need to know Tcrit • Tcrit is defined as the value that marks the most extreme 5% (usually) of the distribution • 5% when  = 0.05 • Different distributions are different shapes and so will have different critical values for the extreme 5% of scores • So, when performing a t-test, we use one specific curve (and one set of critical values) depending on the value of df (or, N-1)

The T-distribution (Tcrit) • Example: Assume the experiment had N=22 and  = 0.05, and we want a two-tailed test • df = N-1 = 22-1 = 21 • Look up t-tables (page 551 of book) • df  = 0.05  = 0.01 • 1 12.706 63.657 • 2 4.303 9.925 • 3 3.182 5.841 • 21 2.080 2.831

The T-distribution (Tcrit) • Practice: What are the Tcrit values for each of the following scenarios • N=16;  = 0.05; Two-tailed • N=31;  = 0.05; Two-tailed • N=28;  = 0.01; Two-tailed • N=9;  = 0.05; Two-tailed • N=25;  = 0.05; One-tailed • N=15;  = 0.01; One-tailed

The T-distribution (Tcrit) • Practice: What are the Tcrit values for each of the following scenarios • N=16;  = 0.05; Two-tailed ±2.131 • N=31;  = 0.05; Two-tailed ±2.042 • N=28;  = 0.01; Two-tailed ±2.771 • N=9;  = 0.05; Two-tailed ±2.306 • N=25;  = 0.05; One-tailed 1.711 • N=15;  = 0.01; One-tailed 2.624

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 • What is the appropriate statistical test? • Is this a one-tailed or two-tailed test? Why? • What are H0 and HA? • Compute Tobt • With =0.05, what is Tcrit? • What conclusion should we draw?

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 1. What is the appropriate statistical test?

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 1. What is the appropriate statistical test? We are comparing a sample of scores to a population mean, therefore we will use a one-sample test. As we do not know the population variance, we must estimate it and use a one-sample T-test

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 1. Is this a one-tailed or two-tailed test? Why?

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 1. Is this a one-tailed or two-tailed test? Why? Two-tailed We are interested in whether men differ from women

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 2. What are H0 and H1?

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 2. What are H0 and H1? H0 : Men and women are equally enthusiastic H0 : men = 5.23 H1 : Men and women differ in enthusiasm H1 : men  5.23

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 3. Compute Tobt

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 3. Compute TobtTobt = (X - µ) / sx sx= √(s2x / N) =5.23; N=7 X= ?? sX= ?? s2X= ??

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 sx= √(s2x / N) 3. Compute Tobt X=(43/7)=6.14 s2X = [273-(1849/7)] / [7-1]=(273-264.14)/6=1.48 sX= √(1.48/7) = √(0.21) = 0.46

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 3. Compute Tobt Tobt = (X - µ) / sx =5.23; N=7; X=6.14; sX= 0.46 Tobt = (6.14 - 5.23) / (0.46) Tobt = 1.97

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 4. With =0.05, what is Tcrit?

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 4. With =0.05, what is Tcrit? =0.05, two-tailed df=N-1=7-1=6 Tcrit = 2.447

Tcrit and Tobt  Tobt=+1.97 Tcrit= -2.447 Tcrit= +2.447 a a a a

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 5. What conclusion should we draw?

Problem 1 Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the  for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class: 5, 7, 5, 7, 8, 6, 5 5. What conclusion should we draw? As Tobs = Tcrit , we retain H0 Men do not differ significantly from women on how enthusiastic they are about this statistics class

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 • What is the appropriate statistical test? Is this a one-tailed or two-tailed test? Why? • What are H0 and HA? • Compute the obtained value • With =0.05, what is the critical value? • What conclusion should we draw from this study?

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 • What is the appropriate statistical test?

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 • What is the appropriate statistical test? We are comparing a sample of scores to a population mean, therefore we will use a one-sample test. As we do not know the population variance, we must estimate it and use a one-sample T-test

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 1. Is this a one-tailed or two-tailed test? Why?

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 1. Is this a one-tailed or two-tailed test? Why? It will be a one-tailed test, as we are predicting the direction that the scores will change. That is, we are specifically asking whether smoking leads to a decreased sense of smell

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 2. What are H0 and H1?

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 2. What are H0 and H1? H0 : Smoking is not associated with decreased sense of smell H0 : smokers >= 18.4 H1 : Smoking is associated with a decreased sense of smell H1 : smokers < 18.4

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 3. Compute the obtained value

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 • Compute the obtained value Tobt = (X - µ) / sx sx= √(s2x / N) =18.4; N=12 X= ?? sX= ?? s2X= ??

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 • Compute the obtained value X=(195/12)=16.25 s2X = [3221-(38025/12)] / [12-1]=(3221-3168.75)/11=4.75 sX= √(4.75/12) = √(0.396) = 0.629

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 • Compute the obtained value Tobt = (X - µ) / sx =18.4; N=12; X=16.25; sX= 0.629 Tobt = (16.25 - 18.4) / (0.629) Tobt = -3.42

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 4. With =0.05, what is the critical value?

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 • With =0.05, what is the critical value? =0.05, one-tailed df=N-1=12-1=11 Tcrit = -1.796

Tcrit and Tobt  Tobt=-3.42 Tcrit=-1.796 a a a a

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 5. What conclusion should we draw from this study?

Problem 2 A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the  for nonsmokers is 18.4. People who smoke a pack a day produced the following scores: 16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14 5. What conclusion should we draw from this study? As Tobs < Tcrit , we reject H0 and acceptH1. People who smoke have a significantly decreased sense of smell.

Psych 230 Psychological Measurement and Statistics