1. Stats 95 t-Tests Single Sample Paired Samples Independent Samples
William Sealy Gosset
Single Sample
Paired Samples
Independent Samples

2. t Distributions Same Three Assumptions
t dist. are used when we know the mean of the population but not the SD of the population from which our sample is drawn
t dist. are useful when we have small samples.
t dist is flatter and has fatter tails
As sample size approaches 30, t looks like z (normal) dist.
Same Three Assumptions
Dependent Variable is scale
Random selection
Normal Distribution

3. Fat Tails Lose Weight With Larger Sample Size

4. The Robust Nature of the t Statistics
Unfortunately, we very seldom know the if the population is normal because usually all the information we have about a population is in our study, a sample of Fortunately, 1) distributions in social sciences often approximate a normal curve, and 2) according to Central Limit Theorem the sample mean you have gathered is part of a normal distribution of sample means, and 3) in practice t tests statisticians have found the test is accurate even with populations far from normal

5. The Robust Nature of the t Statistics
The only situation in which using a t test is likely to give a seriously distorted result is when you are using a one-tailed test and the population is highly skewed.

6. When To Use Single Sample t-Statistic
1 Nominal Independent Variable
1 Scale Dependent Variable
Population Mean, No Population Standard Deviation
Check Assumptions
Random Selection
Normal Shape Distribution

7. z Statistic Versus t Statistic
When you know the Mean and Standard deviation of a population.
E.g., a farmer picks 200,000 apples, the mean weight is 112 grams, the SD is 12grams.
Calculate the Standard Error of the sample mean
When you do not know the Mean and Standard Deviation of the population
E.g., a farmer picks 30 out of his 200,000 apples, and finds the sample has a Mean of 112 grams.
Calculate the Estimate of the Standard Error of the sample mean

8. Scenarios When you would use a Single Sample t test
A newspaper article reported that the typical American family spent an average of $81 for Halloween candy and costumes last year. A sample of N = 16 families this year reported spending a mean of M = $85, with s = $20. What statistical test would we use to determine whether these data indicate a significant change in holiday spending?
Many companies that manufacture lightbulbs advertise their 60-watt bulbs as having an average life of 1000 hours. A cynical consumer bought 30 bulbs and burned them until they failed. He found that they burned for an average of M = 1233, with a standard deviation of s = What statistical test would this consumer use to determine whether the average burn time of lightbulbs differs significantly from that advertised?

9. Difference Between Calculating z Statistic and t Statistic
Standard Deviation of a Sample: Estimates the Population Standard Deviation
Standard Error of a Sample: estimates the Sample Error of the Population
t Statistic for Single-Sample t Test

10. Estimating Population from a Sample
Main difference between t Tests and z score:
use the standard deviation of the sample to estimate the standard deviation of the population.
How? Subtract 1 from sample size! (called degrees of freedom)
Use degrees of freedom (df) in the t distribution chart
Standard Deviation of a Sample: Estimates the Population Standard Deviation

11. t Distribution Table

12. Example of Single Sample t Test
The mean emission of all engines of a new design needs to be below 20ppm if the design is to meet new emission requirements. Ten engines are manufactured for testing purposes, and the emission level of each is determined. Data:
15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9, 12.7, 13.9
Does the data supply sufficient evidence to conclude that type of engine meets the new standard, assuming we are willing to risk a Type I error (false alarm, reject the Null when it is true) with a probability = 0.01?
Step 1: Assumptions: dependent variable is scale, Randomization, Normal Distribution
Step 2: State H0 and H1:
H0 Emissions are equal to (or lesser than) 20ppm;
H1 Emissions are greater than 20ppm (One-Tailed Test)

13. Example of Single Sample t Test
The mean emission of all engines of a new design needs to be below 20ppm if the design is to meet new emission requirements. Ten engines are manufactured for testing purposes, and the emission level of each is determined. Data:
15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9, 12.7, 13.9
Step 3: Determine Characteristics of Sample
Mean =
Standard Deviation of Sample =
Standard Error of Sample =
Step 4: Determine Cutoff
df = N-1 = 10-1 =9
t statistic cut-off =

14. Example of Single Sample t Test
The mean emission of all engines of a new design needs to be below 20ppm if the design is to meet new emission requirements. Ten engines are manufactured for testing purposes, and the emission level of each is determined. Data:
15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9, 12.7, 13.9
Step 3: Determine Characteristics of Sample
Mean M =
Standard Deviation of Sample s =
Standard Error of Sample sm = 0.942
Step 4: Determine Cutoff
df = N-1 = 10-1 =9
t statistic cut-off =
Step 5: Calculate t Statistic

15. Example of Single Sample t Test
The mean emission of all engines of a new design needs to be below 20ppm if the design is to meet new emission requirements. Ten engines are manufactured for testing purposes, and the emission level of each is determined. Data:
15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9, 12.7, 13.9
Mean M = Standard Deviation of Sample s = Standard Error of Sample sm = 0.942
Step 5: Calculate t Statistic
Step 6: Decide (Draw It)
t statistic cut-off =
t statistic = -3.00
Decide to reject the Null Hypothesis

16. How to Write Results t(7) = -.79, p < .265, d = -.29
t Indicates that we are using a t-Test
(9) Indicates the degrees of freedom associated with this t-Test
-.79 Indicates the obtained t statistic value
p < .265 Indicates the probability of obtaining the given t value by chance alone
d = -.29 Indicates the effect size for the significant effect (the magnitude of the effect is measured in standard deviation units)

17. Paired Sample t Test
The paired samples test is a kind of research called repeated measures test (aka, within-subjects design), commonly used in before-after-designs.
Comparing a mean of difference scores to a distribution of means of difference scores
Checklist for Paired-Samples:
1 Nominal DICHOTOMOUS (with two levels) Independent Variable
1 Scale Dependent Variable
Paired Observations are Dependent
Assumptions
Random selection & Shape

18. Paired-Samples t-Test

19. Paired Sample t Test
The paired samples test is a kind of research called repeated measures test (aka, within-subjects design), commonly used in before-after-designs.
Comparing a mean of difference scores to a distribution of means of difference scores
Population of measures at Time 1 and Time 2
Population of difference between measures at Time 1 and Time 2
Population of mean difference between measures at Time 1 and Time 2
(Whew!)

20. Paired Sample t Test Single-Sample Paired-Sample
Single observation from each participant
The observation is independent from that of the other participants
Comparing a mean score to a distribution of mean scores .
Two observations from each participant
The second observation is dependent upon the first since they come from the same person.
Comparing a mean of difference scores to a distribution of means of difference scores
(I don’t make this stuff up)

21. Paired Sample t Test A distribution of differences between scores.
Two distribution of scores.
Central Limit Theorem Revisited. If you plot the mean of randomly sampled observations, the plot will approach a normal distribution. This is true for scores and for differences between matched scores.

22. Difference Between Calculating Single-Sample t and Paired-Sample t Statistic
Single Sample t Statistic
Paired Sample t Statistic
Standard Deviation of a Sample
Standard Deviation of Sample Differences
Standard Error of a Sample
Standard Error of Sample Differences
t Statistic for Single-Sample t Test
t Statistic for Paired-Sample t Test

23. Paired Sample t Test Example
We need to know if there is a difference in the salary for the same job in Boise, ID, and LA, CA. The salary of 6 employees in the 25th percentile in the two cities is given .
Six Steps of Hypothesis testing for Paired Sample Test
Profession
Boise
Los Angeles
Executive Chef
53,047
62,490
Genetics Counselor
49,958
58,850
Grants Writer
41,974
49,445
Librarian
44,366
52,263
School teacher
40,470
47,674
Social Worker
36,963
43,542

24. Paired Sample t Test Example
We need to know if there is a difference in the salary for the same job in Boise, ID, and LA, CA.
Step 1: Define Pops. Distribution and Comparison Distribution and Assumptions
Pop. 1. Jobs in Boise
Pop. 2.. Jobs in LA
Comparison distribution will be a distribution of mean differences, it will be a paired-samples test because every job sampled contributes two scores, one in each condition.
Assumptions: the dependent variable is scale, we do not know if the distribution is normal, we must proceed with caution; the jobs are not randomly selected, so we must proceed with caution

25. Paired Sample t Test Example
We need to know if there is a difference in the salary for the same job in Boise, ID, and LA, CA.
Step 3: Determine the Characteristics of Comparison Distribution (mean, standard deviation, standard error)
M = Sum of Squares (SS) = 5,777,
Profession
Boise
Los Angeles
X-Y
D (X-Y)-M
M =
D^2
Executive Chef
53,047
62,490
-9,443
-1,528.67
2,336,821.78
Genetic Counselor
49,958
58,850
-8,892
955,832.11
Grants Writer
41,974
49,445
-7,471
443.33
196,544.44
Librarian
44,366
52,263
-7,897
17.33
300.44
School teacher
40,470
47,674
-7,204
710.33
504,573.44
Social Worker
36,963
43,542
-6,579
1,335.33
1,783,115.11

26. Paired Sample t Test Example
We need to know if there is a difference in the salary for the same job in Boise, ID, and LA, CA.
Step 4: Determine Critical Cutoff
df = N-1 = 6-1= 5
t statistic for 5 df , p < .05, two-tailed, are and 2.571
Step 5: Calculate t Statistic
Step 6 Decide

27. How to Write Results t(5) = 18.04, p < .00001, d = xxxx.
t Indicates that we are using a t-Test
(5) Indicates the degrees of freedom associated with this t-Test
18.04 Indicates the obtained t statistic value
p < Indicates the probability of obtaining the given t value by chance alone
d = xxxx Indicates the effect size for the significant effect (the magnitude of the effect is measured in units of standard deviations)

28. Independent t-Test

29. Independent t Test
Compares the difference between two means of two independent groups.
You are comparing a difference between means to a distribution of differences between means.
Sample means from Group 1 and Group 2 compared to a Population of differences between means of Group 1 and Group 2

30. Independent t Test Paired-Sample Independent t Test
Two observations from each participant
The second observation is dependent upon the first since they come from the same person.
Comparing a mean difference to a distribution of mean difference scores
Single observation from each participant from two independent groups
The observation from the second group is independent from the first since they come from different subjects.
Comparing the difference between two means to a distribution of differences between mean scores .

31. Independent t Test: Steps
Calculate the corrected variance for each sample
Step 1
Calculate the degrees of freedom
Step 2
Calculate the pooled variance
Step 3
In a z test you find SD, for ind. t, you take weighted avg of SD2
Calculate the squared estimate of the standard error for each sample
Like the Standard error in a z test
Step 4
Step 5
Calculate estimated variance of the dist. of mean differences

32. Independent t Test: Steps
Calculate the squared estimate of the standard error for each sample
Step 4
Calculate estimated variance of the dist. of mean differences
Step 5
Calculate estimated standard deviation of the dist. of mean differences
Step 6
t Statistic for an independent-Samples t Test
Step 7

33. Confidence Interval for Independent Samples t-test
Confidence Interval for Paired Sample t-test with example from Stroop question.
Effect Size for Dependent Samples t-test
Effect Size for Independent Samples t-test

34. Standard Error of a Sample: estimates the Sample Error of the Population
t Statistic for Single-Sample t Test
Standard Deviation of a Sample: Estimates the Population Standard Deviation
t Statistic for Independent t Test
Degrees of Freedom for Single Sample t Test, and Paired Sample
Variance for a sample
T Statistic for Paired-Sample t Test

35. Degrees of Freedom for Independent Samples t Test
Pooled Variance. Like adding together the weighted average of the variance from Variable X and Variable Y.
Variance for a Distribution of means for Indep.-Samples t Test
Variance for a Distribution of Differences between Means
SD of the distribution of Differences Between Means

39. Central Limit Theorem in Single Independent T-tests: Samples
Two samples (moms/non-moms; drunk /sober students etc.) independent of each other (subjects participate in only one condition) which you draw from a universe of respective possible samples.

40. Central Limit Theorem in Single Independent t-tests: Distribution of Samples
Two samples (moms/non-moms; drunk /sober students etc.) independent of each other (subjects participate in only one condition) is drawn from a hypothetical distribution of samples of the same size, with mean of samples and a standard error.

41. Central Limit Theorem in Single Independent t-tests: Distribution of Samples
Difference between the means. Δ Mx-My
Difference between the means. Would it occur with a probability of 5% or less?
We want to know if there is a statistically significant difference between the means.

42. Central Limit Theorem in Single Independent t-tests: Distribution of Differences Between Means
Difference between the means, Δ Mx-My, comes from a distribution of Δ Mx-My that has a standard deviation drawn from the pooled variance of the sample distributions
We want to know if there is a statistically significant difference between the means.

43. Central Limit Theorem in Single Independent T-tests