1. Hypothesis Testing in the Real World
Thus far, Hypothesis testing has been presented for situations in which the population variances are known.
T-tests allow you to compare means for which the population variances are unknown
T-test for a single sample

2. Hypothesis Testing in the Real World
T-test for a single sample
This involves the comparison of a single sample to a population of known mean but unknown variance
Example: Do students in my fraternity study more hours than the national average of 2.5 hours per day.
I conduct a survey of 20 men and discover that they study an average of 3.2 hours per day.
Is 3.2 significantly greater than 2.5?

3. Hypothesis Testing in the Real World
Steps of Hypothesis Testing
Step 1: Restate the research question as a research hypothesis and a null hypothesis - refer to populations
Population 1: Students who live in fraternities
Population 2: Students who live in the general college population
Research Hypothesis: The mean study time of Population 1 will be higher than the mean study time of Population 2
Null Hypothesis: The mean study time of Population 1 will be the same as the mean study time of Population 2

4. Hypothesis Testing in the Real World
Steps of Hypothesis Testing
Step 2: Determine the comparison distribution
The mean of Population 2 (All students; 2.5) is the mean of the comparison distribution
The variance of Population 2 is unknown
Estimate the population variance from the sample variance
Since the variance of a sample will be slightly smaller than the variance of a population, the estimate based on a sample is biased.
You can calculate an unbiased estimate by using a slightly different formula for calculating the variance.

5. Hypothesis Testing in the Real World
Step 2: Determine the comparison distribution
You can calculate an unbiased estimate of the population variance by using a slightly different formula for calculating the variance.
Variance=Sum (X-M) squared, divided by N
Unbiased Variance=Sum (X-M) squared, divided by N-1. Division by N-1 makes the variance estimate larger.
The number you divided by (N-1) is the “Degrees of Freedom” (df). It is the number of scores in the sample that are “free to vary”. If you know the mean of a sample, then one score is determined and not free to vary.
Unbiased Variance=SS/df

6. Hypothesis Testing in the Real World
Step 2: Determine the comparison distribution
Calculate the Standard Deviation of the Distribution of Means
Variance of Means = Population Variance / N
= 2.54 / 12 = .21
SD of Means = Square Root of Variance of Means
= .46

7. Step 2: Determine the comparison distribution
The t distribution is the shape of the comparison distribution when you use an estimated population variance.
When you use an estimate of the population variance that is derived from a sample, there is more error. This results in more extreme means in the distribution of means. The smaller the sample, the more this occurs.

8. Step 2: Determine the comparison distribution
When you use an estimate of the population variance that is derived from a sample, there is more error. This results in more extreme means in the distribution of means. The smaller the sample, the more this occurs.
You have a slightly different t distribution for every different degrees of freedom (df)

9. Step 2: Determine the comparison distribution
When you use an estimate of the population variance that is derived from a sample, there is more error. This results in more extreme means in the distribution of means. The smaller the sample, the more this occurs.
The difference in shape affects how extreme a score must be in order to reject the null hypothesis. You need a slightly more extreme mean to reject the null with a t distribution than with a normal distribution.

10. Step 2: Determine the comparison distribution
When you use an estimate of the population variance that is derived from a sample, there is more error. This results in more extreme means in the distribution of means. The smaller the sample, the more this occurs.
The greater the df, the more the t distribution approximates the normal curve. Your greater sample size results in a more accurate estimate of the population variance.

11. Step 2: Determine the comparison distribution
When you use an estimate of the population variance that is derived from a sample, there is more error.
Example
.05 alpha for normal distribution = 1.64
.05 alpha for t distribution w 7 df = 1.89
.05 alpha for t distribution w 25 df = 1.71
.05 alpha for t distribution w infinite df = 1.64

12. Step 3: Determine the cut-off on the comparison distribution for rejecting the null hypothesis
There is a different t distribution for every df. Most t tables only include the cut-off points for common alpha levels.

13. Step 3: Determine the cut-off on the comparison distribution for rejecting the null hypothesis

14. Step 4: Figure your sample’s score on the comparison distribution
When calculating this before, we rendered a Z score. When comparing to a t distribution, you calculate a t-score.
T = (Sample Mean - the population mean) divided by the standard deviation of the distribution of means
Example
Sample mean = 4
Population mean = 2.5
SD of the means = .46
T= / .46
T= 3.26

15. Step 5: Decide whether to reject the null hypothesis

16. Step 5: Decide whether to reject the null hypothesis

17. Hypothesis Testing in the Real World
Example

18. T test for dependent means
Used when you measure a dependent variable twice from each unit of observation (person)
Scores on a reading test before and after phonics training
Ratings on the safety of flying vs. the safety of taking a train, when collected from the same people
Ratings on the taste of Pepsi vs. Coke, when collected from the same people
Speed before and after seeing a police check point

19. T test for dependent means
Other names:
repeated measures design, t-test for paired samples, t-test for correlated means, t-test for matched samples
With a t-test for dependent means, 1) you use a difference score for the comparison rather than a single score and 2) you assume the population mean is 0.

20. T test for dependent means
Example: Speed before and after seeing a police check point. Was there a significant change in speed?
A policeman clocks the following times for a sample of cars approaching a speed checkpoint on a highway with a posted speed limit of 35mph:
30, 36, 42, 36, 30, 52, 36, 34, 36, 33, 30, 32, 35, 32, 37.
A policeman clocks the following times for a sample of cars leaving a speed checkpoint on a highway with a posted speed limit of 35mph:
31, 37, 32, 34, 32, 40, 31, 35, 35, 32, 31, 32, 33, 33, 32.

21. T test for dependent means
Example Step 1: Restate the question as a research hypothesis and null hypothesis
Population 1: People who see a speed check-point
Population 2: People whose speed does not change after seeing a speed check-point
Research Hypothesis: People do change speed after seeing a speed check-point.
Null Hypothesis: Speed stays the same before and after seeing a speed check-point.

22. T test for dependent means
Example Step 2: Determine the characteristics of the comparison distribution
If the null hypothesis is true then the mean of the difference scores will be 0.
The variance of the population of difference scores can be estimated from the sample of difference scores

23. T test for dependent means
Example Step 3: Determine the cutoff score on the comparison distribution
A two tailed test for alpha <.05, df = 14 has a cutoff t-score of 2.145

24. T test for dependent means
Example Step 4: Determine the t-score for your study

25. T test for dependent means
Example Step 5: Decide whether to reject the null hypothesis
Is a t-score of -1.9 greater than the cutoff of ± 2.145?
Now, let’s compare these results for the case of the single mean and the dependent means to those using SPSS.

26. T-test for Dependent Means
Now, let’s compare these results for the case of the single mean and the dependent means to those using SPSS.
Navigate to Speed Trap Data at learnpsychology.com

27. T-test for Dependent Means
Another T-test for dependent means example:
Statistical Parameter Mapping for Functional Magnetic Resonance Imaging (fMRI).
Step 1: Record resonant radio signals from tissue responding to radio pulses within a strong magnetic field.

28. T-test for Dependent Means
Statistical Parameter Mapping for Functional Magnetic Resonance Imaging (fMRI).
Step 2: Map the signal strengths as gray levels. This renders a map of different tissues.

29. T-test for Dependent Means
Statistical Parameter Mapping for Functional Magnetic Resonance Imaging (fMRI).
These maps are made pixel by pixel.

30. T-test for Dependent Means
Statistical Parameter Mapping for Functional Magnetic Resonance Imaging (fMRI).
The BOLD Response

31. T-test for Dependent Means
Statistical Parameter Mapping for Functional Magnetic Resonance Imaging (fMRI).
The “Boxcar” Design. Design A has greater variance than Design B.
T = 2.3
T = 6.7

32. T-test for Dependent Means
Statistical Parameter Mapping for Functional Magnetic Resonance Imaging (fMRI).
Calculate t scores for every pixel and then plot these as colored areas
This is a Statistical Parameter Map (SPM).
This image depicts the areas of elevated signal associated with hearing spoken words.

33. T-test for Dependent Means
Statistical Parameter Mapping for Functional Magnetic Resonance Imaging (fMRI).
Sample application: Presurgery planning

34. T test for dependent means
Assumptions of the t-test
The distribution of individuals must follow a normal curve. The comparison distribution (t-distribution) is derived from the distribution of individuals. If this distribution is not normal then the t distribution has an abnormal shape.

35. T test for dependent means
Effect Size and the t-test for dependent means
d = population 1 mean - population 2 mean / variance
For the t-test for dependent means, the mean of population 1 is 0, the mean of the difference scores.
The variance is usually the standard deviation of the difference scores
What was the effect size for our example?
= / 4.2
= .49
Effect is small to moderate

36. T test for dependent means
Power and the t-test for dependent means
Effect Size
P<.05 N
Small
Medium
Large
One tail
D=.20
D=.5
D=.8 10
.15
.46
.78 20
.22
.71
.96 30
.29
.86
.99 40
.35
.93 50
.40
.97
100
.63

37. T test for dependent means
Estimating your sample size in relationship to power
Effect Size
P<.05
Small
Medium
Large
d=.2
d=.5
d=.8
One tailed
156 26 12
Two tailed
196 33 14

38. Homework Null H True Research H True
Research H Supported Reject Null H
Type 1 Error: Study concludes that premature infants are different in face recognition when they are the same as full-term infants
Study correctly concludes that premature infants are different in facial recognition
Do not reject Null H, Study inconclusive
Study correctly concludes that the results are inconclusive and the null hypothesis is not rejected, research hypothesis is not supported
Type 2 Error: Study concludes that results are inconclusive and research hypothesis is not supported when in fact there is a difference between the infants