1 / 77

Psych 230 Psychological Measurement and Statistics

Psych 230 Psychological 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.

judith
Download Presentation

Psych 230 Psychological Measurement and Statistics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

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

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

  4. 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

  5. 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

  6. 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

  7. 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

  8. One sample T-test

  9. 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

  10. 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

  11. 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

  12. 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

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

  14. 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

  15. 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)

  16. 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

  17. 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

  18. 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

  19. 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?

  20. 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?

  21. 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

  22. 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?

  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 1. Is this a one-tailed or two-tailed test? Why? Two-tailed We are interested in whether men differ from women

  24. 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?

  25. 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

  26. 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

  27. 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= ??

  28. 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

  29. 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

  30. 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?

  31. 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

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

  33. 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?

  34. 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

  35. 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?

  36. 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?

  37. 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

  38. 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?

  39. 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

  40. 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?

  41. 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

  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 3. Compute the obtained value

  43. 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= ??

  44. 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

  45. 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

  46. 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?

  47. 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

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

  49. 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?

  50. 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.

More Related