1. t-Tests

2. Interval Estimation and the t Distribution

3. Large Sample z-Test
Sometimes we have reason to test hypotheses involving specific values for the mean.
Example 1. Claim: On average, people sleep less than the often recommended eight hours per night.
Example 2. Claim: On average, people drink more than the recommended 2 drinks per day.
Example 3. Claim: On average, women take more than 4 hours to run the marathon.
However, it is rare that we have a specific hypothesis about the standard deviation of the population under study.
For these situations, we can use the sample standard deviation s as an estimator for the population standard deviation s.
If the sample size is pretty big (e.g., >100), then this estimate is pretty good, and we can just use the standard z test.
PSYC 6130, PROF. J. ELDER

4. Example: Canadian General Social Survey, Cycle 6 (1991)
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5. But what if we don’t have such a large sample?
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6. Student’s t Distribution
Problem: for small n, s is not a very accurate estimator of s.
The result is that the computed z-score will not follow a standard normal distribution.
Instead, the standardized score will follow what has become known as the Student’s t distribution.
PSYC 6130, PROF. J. ELDER

7. Student’s t Distribution
Normal distribution
t distribution, n=2, df=1
t distribution, n=10, df=9
t distribution, n=30, df=29
How would you describe the difference between the normal and t distributions?
PSYC 6130, PROF. J. ELDER

8. Student’s t distribution
Student’s t distribution is leptokurtic
More peaked
Fatter tails
What would happen if we were to ignore this difference, and use the standard normal table for small samples?
PSYC 6130, PROF. J. ELDER

9. Student’s t Distribution
Critical t values decrease as df increases
As df  infinity, critical t values  critical z values
Using the standard normal table for small samples would result in an inflated rate of Type I errors.
PSYC 6130, PROF. J. ELDER

10. One-Sample t Test: Example
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11. PSYC 6130, PROF. J. ELDER

12. Reporting Results
Respondents who report being very forgetful sleep, on average, 7.11 hours/night, significantly less than the recommended 8 hours/night, t(37)=2.25, p<.05, two-tailed.
PSYC 6130, PROF. J. ELDER

13. Confidence Intervals
NHT allows us to test specific hypotheses about the mean.
e.g., is m < 8 hours?
Sometimes it is just as valuable, or more valuable, to know the range of plausible values.
This range of plausible values is called a confidence interval.
PSYC 6130, PROF. J. ELDER

14. Confidence Intervals
The confidence interval (CI) of the mean is the interval of values, centred on the sample mean, that contains the population mean with specified probability.
e.g., there is a 95% chance that the 95% confidence interval contains the population mean.
NB: This assumes a flat prior on the population mean (non-Bayesian).
Confidence Interval
PSYC 6130, PROF. J. ELDER

15. Confidence Intervals
95% Confidence Interval
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16. Basic Procedure for Confidence Interval Estimation
Select the sample size (e.g., n = 38)
Select the level of confidence (e.g., 95%)
Select the sample and collect the data (Random sampling!)
Calculate the limits of the interval
PSYC 6130, PROF. J. ELDER

17. End of Lecture 4
Oct 8, 2008

18. Selecting Sample Size Suppose that
You have a rough estimate s of the standard deviation of the population, and
You want to do an experiment to estimate the mean within some 95% confidence interval of size W.
PSYC 6130, PROF. J. ELDER

19. Assumptions Underlying Use of the t Distribution for NHT and Interval Estimation
Same as for z test:
Random sampling
Variable is normal
CLT: Deviations from normality ok as long as sample is large.
Dispersion of sampled population is the same as for the comparison population
PSYC 6130, PROF. J. ELDER

20. Sampling Distribution of the Variance

21. Sampling Distribution of the Variance
We are sometimes interested in testing a hypothesis about the variance of a population.
e.g., is IQ more diverse in university students than in the general population?
PSYC 6130, PROF. J. ELDER

22. Sampling Distribution of the Variance
What form does the sampling distribution of the variance assume?
If the variable of interest (e.g., IQ) is normal, the sampling distribution of the variance takes the shape of a c-squared distribution:
p(s2) s2
PSYC 6130, PROF. J. ELDER

23. Sample Variances and the c-Square Distribution50
100
150
n=9
n=29
n=99
p(c2) c2
PSYC 6130, PROF. J. ELDER

24. Sample Variances and the c-Square Distribution
The c-square distribution is:
strictly positive.
positively skewed.
Since the sample variance is an unbiased estimator of the population variance: E(s2) = s 2
Due to the positive skew, the mean of the distribution E(s2) is greater than the mode.
As the sample size increases, the distribution approaches a normal distribution.
If the original distribution is not normal and the sample size is not large, the sampling distribution of the variance may be far from c-square, and tests based on this assumption may be flawed.
PSYC 6130, PROF. J. ELDER

25. Example: Height of Female Psychology Graduate Students
2005 PSYC 6130A Students (Female)
Source: Canadian Community
Health Survey Cycle 3.1 (2005)
Caution: self report!
PSYC 6130, PROF. J. ELDER

26. Properties of Estimators
We have now met two statistical estimators:
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27. NHT for Two Independent Sample Means

28. Conditions of Applicability
Comparing two samples (treated differently)
Don’t know means of either population
Don’t know variances of either population
Samples are independent of each other
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29. Example: Height of Canadian Males by Income Category (Canadian Community Health Survey, 2004)
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30. Sampling Distribution
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31. Sampling Distribution (cntd…)
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32. NHT for Two Large Samples
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33. Height of Canadian Males by Income Category (Canadian Community Health Survey, 2004)
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34. NHT for Two Small Samples

35. Example: Social Factors in Psychological Well-Being
Canadian Community Health Survey, 2004
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36. Social Factors in Psychological Well-Being (cntd…)
Canadian Community Health Survey, 2004
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37. Social Factors in Psychological Well-Being (cntd…)
Canadian Community Health Survey, 2004:
Respondents who report never getting along with others
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38. NHT for Two Small Independent Samples
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39. NHT for Two Small Independent Samples (cntd…)
PSYC 6130, PROF. J. ELDER

40. Pooled Variance
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41. Social Factors in Psychological Well-Being (cntd…)
Canadian Community Health Survey, 2004:
Respondents who report never getting along with others
PSYC 6130, PROF. J. ELDER

42. Reporting the Result
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43. Confidence Intervals for the Difference Between Two Means
PSYC 6130, PROF. J. ELDER

44. Underlying Assumptions
Dependent variable measured on interval or ratio scale.
Independent random sampling
(independence within and between samples)
In experimental work, often make do with random assignment.
Normal distributions
Moderate deviations ok due to CLT.
Homogeneity of Variance
Only critical when sample sizes are small and different.
PSYC 6130, PROF. J. ELDER

45. End of Lecture 5
Oct 15, 2008

46. Social Factors in Psychological Well-Being (cntd…)
Canadian Community Health Survey, 2004:
Respondents who report never getting along with others
PSYC 6130, PROF. J. ELDER

47. Separate Variances t TestIf
Population variances are different (suggested by substantially different sample variances)
AND
Samples are small
Sample sizes are substantially different
Then
Pooled variance t statistic will not be correct.
In this case, use separate variances t test
PSYC 6130, PROF. J. ELDER

48. Separate Variances t Test
This statistic is well-approximated by a t distribution.
Unfortunately, calculating the appropriate df is difficult.
SPSS will calculate the Welch-Satterthwaite approximation for df as part of a 2-sample t test:
PSYC 6130, PROF. J. ELDER

49. Social Factors in Psychological Well-Being (cntd…)
Canadian Community Health Survey, 2004:
Respondents who report never getting along with others
PSYC 6130, PROF. J. ELDER

50. Summary: t-Tests for 2 Independent Sample Means
PSYC 6130, PROF. J. ELDER

51. More on Homogeneity of Variance
How do we decide if two sample variances are different enough to suggest different population variances?
Need NHT for homogeneity of variance.
F-test
Straightforward
Sensitive to deviations from normality
Levene’s test
More robust to deviations from normality
Computed by SPSS
PSYC 6130, PROF. J. ELDER

52. Levene’s Test: Basic Idea
SPSS reports an F-statistic for Levene’s test
Allows the homogeneity of variance for two or more variables to be tested.
We will introduce the F distribution later in the term.
PSYC 6130, PROF. J. ELDER

53. The Matched t Test

54. Independent or Matched?
Application of the Independent-Groups t test depended on independence both within and between groups.
There are many cases where it is wise, convenient or necessary to use a matched design, in which there is a 1:1 correspondence between scores in the two samples.
In this case, you cannot assume independence between samples!
Examples:
Repeated-subject designs (same subjects in both samples).
Matched-pairs designs (attempt to match possibly important attributes of subjects in two samples)
PSYC 6130, PROF. J. ELDER

55. Example: Assignment Marks
These scores are not independent!
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56. Better alternative: The matched t-test using the direct difference method
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57. Matched vs Independent t-test
Why does a matched t-test yield a higher t-score than an independent t-test in this example?
The t-score is determined by the ratio of the difference between the groups and the variance within the groups.
The matched t-test factors out the portion of the within-group variance due to differences between individuals.
PSYC 6130, PROF. J. ELDER

58. The Matched t Test and Linear Correlation
The degree to which the matched t value exceeds the independent-groups t value depends on how highly correlated the two samples are.
Alternate formula for matched standard error:
PSYC 6130, PROF. J. ELDER

59. Case 1: r = 0 Independent t-test Matched t-test
Thus the t-score will be the same.
But note that
Thus the critical t-values will be larger for the matched test.
PSYC 6130, PROF. J. ELDER

60. Case 2: r > 0 Independent t-test Matched t-test
Now the t-score will be larger for the matched test. Although the critical t-values are larger, the net result is that the matched test will often be more powerful.
PSYC 6130, PROF. J. ELDER

61. Confidence Intervals Just as for one-sample t test:
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62. Repeated Measures Designs
Many matched sample designs involve repeated measures of the same individuals.
This can result in carry-over effects, including learning and fatigue.
These effects can be minimized by counter-balancing the ordering of conditions across participants.
PSYC 6130, PROF. J. ELDER

63. Assumptions of the Matched t Test
Normality
Independent random sampling (within samples)
PSYC 6130, PROF. J. ELDER