1. Comparing Means from Two Data Sets
The t-test
6/16/2019
HK Dr. Sasho MacKenzie

2. Research Questions
To improve muscular power, should an athlete perform heavy resistance exercises, or light plyometric exercises?
Is it better to imagine the flight of the ball or the actions of your swing prior to striking a golf ball?
Is running 5 km or walking 5 km better for burning calories?
Do golfers sink more putts if they focus on the hole or on the ball during a putt?
Will squatting to a lower depth during a vertical jump improve performance?
6/16/2019
HK Dr. Sasho MacKenzie

3. The t-test
All of the questions posed on the previous slide can be statistically addressed using the t-test.
A t-test determines if two groups of data are significantly different (not meaningfully different).
A t-test is the ratio of the actual difference between two means to the difference that is expected due to chance alone.
The bigger the actual difference is compared to the expected difference due to chance, the more statistically significant the t-test.
6/16/2019
HK Dr. Sasho MacKenzie

4. The t-test
A t-test calculation produces a value (t-statistic) that is similar to a z-score.
The t-distributions, are also very similar to the z-score distribution (normal distribution).
The t-distribution changes depending on the sample size.
6/16/2019
HK Dr. Sasho MacKenzie

5. The t Distributions (3 examples)
N = 60 (same as normal curve)
N = 10
N = 3
E.g., Area beyond t=3 increases as N decreases -4 -3 -2 -1 1 2 3 4 t
6/16/2019
HK Dr. Sasho MacKenzie

6. Let’s use an Example
Question: Do HK students drink more or less alcohol than the average St.FX student?
Assumptions:
Every student on campus honestly completed a form and the average drinks/week is known.
Therefore, we know the mean of the population.
Methods:
Determine the drinks/week for a sample of HK students.
Determine if the sample mean is “different” than the population mean (perform a t-test).
6/16/2019
HK Dr. Sasho MacKenzie

7. What is “different”?
Before the t-test, we must set a standard for statistical significance.
This means determining the chance of error we are willing to have in our final decision.
I.e., How confident do we want to be in our decision that HK students drink a different amount?
This decision is represented by alpha (), which is typically set at .05 (5%). This value is arbitrary.
Assume no difference and that the study is repeated 100 times. On 5 occasions, due to chance, we would incorrectly find that HK students drink more.
6/16/2019
HK Dr. Sasho MacKenzie

8. Expected difference due to chance
One-sample t-test
We will use what’s called a one-sample t-test.
This compares the mean of a sample to the mean of a population.
Actual difference
Expected difference due to chance
X = sample mean
 = population mean
SEM = standard error of the mean
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HK Dr. Sasho MacKenzie

9. The Hypothesis
In statistics you must clearly state a testable hypothesis.
Typically the hypothesis tested is opposite to what you expect and is referred to as the null hypothesis.
Our null hypothesis is that HK students do not drink a different amount than the average university student.
X =  or X -  = 0
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HK Dr. Sasho MacKenzie

10. The Calculation University Population HK Sample of students
Average drinks per week = 10
HK Sample of students
Mean = 12, SD = 5, N = 30
The odds of getting a t stat this big, or bigger, due to chance would then be determined by calculating a p-value.
6/16/2019
HK Dr. Sasho MacKenzie

11. The P-value
In Excel, the function TDIST() can be used to calculate the p-value.
The degrees of freedom are N-1.
Our example is a two-tailed test because HK students may drink more, or less, than the average. I.e., the sample mean could be either more or less than the population.
Since we set alpha = .05, if the p-value is less than .05, we will state HK students are statistically different.
6/16/2019
HK Dr. Sasho MacKenzie

12. Graphic of two-tailed one sample t-test
t distribution for N = 30
From TDIST, p = .037
Combined area beyond t=2.19 and t = is .037 -4 -3 -2 -1 1 2 3 4 t
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HK Dr. Sasho MacKenzie

13. Conclusion
Since p=.037 is less than alpha = .05, we reject the null hypothesis and conclude that HK students consume significantly more drinks per week.
The following shows how this would be explained in a study.
It was determined that the average number of alcoholic drinks consumed by HK students (12 drinks), per week, was significantly more than the typical university student (10 drinks), t(29) = 2.19, p = .037.
6/16/2019
HK Dr. Sasho MacKenzie

14. Independent t-test
Determines if two sample means are statistically different.
The null hypothesis is that the means come from the same population, X1 -X2 = 0.
The bottom part of the t-stat now reflects the SEM for both samples, but is still a measure of how much you could expect the means of two samples from the same population to differ due to chance.
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HK Dr. Sasho MacKenzie

15. The Equation
The stuff on the bottom of the equation is called the standard error of the difference.
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HK Dr. Sasho MacKenzie

16. Independent t-test example
Do HK students drink more or less than Chemistry students?
Null Hypothesis: HK students and Chemistry students drink the same amount of alcohol per week.
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HK Dr. Sasho MacKenzie

17. The Calculation HK sample of students Chemistry sample of students
Mean = 12, SD = 5, N = 30
Chemistry sample of students
Mean = 10, SD = 3, N = 30
The odds of getting a t stat this big, or bigger, due to chance would then be determined by calculating a p-value.
6/16/2019
HK Dr. Sasho MacKenzie

18. Graphic of two-tailed independent sample t-test
t distribution for N = 60
From TDIST, p = .065
Combined area beyond t=1.88 and t = is .065 -4 -3 -2 -1 1 2 3 4 t
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HK Dr. Sasho MacKenzie

19. Conclusion
Since p=.065 is greater than alpha = .05, we cannot reject the null hypothesis. There is not enough evidence to suggest HK students drink more or less than Chemistry students
In a study it would be written as:
It was determined that the average number of alcoholic drinks consumed by HK students (12 drinks), per week, was not significantly different than the Chemistry students (10 drinks), t(58) = 1.88, p = .065.
6/16/2019
HK Dr. Sasho MacKenzie

20. Dependent (Paired) t-test
Determines if two correlated sample means are statistically different.
Required when the same subjects are measured twice. E.g., Pre-test, Post-test study.
Adjustments are made in how the variability (SD) in the sample data is calculated. This reduces the denominator in the t-statistic and therefore increases the t-statistic.
This accounts for reduction in the t-statistic due to the fact that the same subjects measured twice will show a smaller mean difference than two completely separate groups.
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HK Dr. Sasho MacKenzie

21. The Difference
Before any group means or standard deviations are calculated, the difference scores between the two measurement times is determined.
For example, if you have a column of pre-test scores and a column of post-test scores, then generate a third column of post minus pre scores.
The t-statistic is then calculated using information from the column of difference scores.
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HK Dr. Sasho MacKenzie

22. The Equation
The variables in this equations come from a single column of difference scores.
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HK Dr. Sasho MacKenzie

23. Dependent t-test example
Do HK students drink more alcohol on the Saturday prior to a Biomechanics midterm, or on the following Saturday?
Null Hypothesis: HK students drink the same amount or less on the following Saturday, compared to the Saturday preceding a Biomechanics midterm.
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HK Dr. Sasho MacKenzie

24. The Data: Number of Drinks
Subject
Sat. Before
Sat. After
Diff 1 16 21
5.0 2 6 9
3.0 3 18 25
7.0 4 11 10
-1.0 5 7
0.0
-3.0 8 12
4.0
2.0 14 15
1.0
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HK Dr. Sasho MacKenzie

25. The Calculation Difference Scores (Before – After)
Mean = 2.1, SD = 3.0, N = 10
The odds of getting a t stat this big, or bigger, due to chance would then be determined by calculating a p-value.
6/16/2019
HK Dr. Sasho MacKenzie

26. The P-value
In Excel, the function TDIST() can be used to calculate the p-value.
The degrees of freedom are (Npairs - 1).
Our example is a one-tailed test because we are assuming HK students drink more following a midterm. This may not be a good assumption, but I needed a one-tailed example.
Set alpha = .05, if the p-value is less than .05, we will state HK students drink significantly more following a biomechanics midterm.
6/16/2019
HK Dr. Sasho MacKenzie

27. Graphic of one-tailed dependent sample t-test
t distribution for N = 10
From TDIST, p = .0258
The area beyond t = 2.24 is .0258 -4 -3 -2 -1 1 2 3 4 t
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HK Dr. Sasho MacKenzie

28. Conclusion
Since p=.0258 is less than alpha = .05, we reject the null hypothesis and conclude that HK students consume significantly more drinks on the Saturday following a midterm.
The following shows how this would be explained in a study.
It was determined that HK students consume significantly more alcoholic drinks (2.1 more) on a Saturday after a midterm than on a Saturday before a midterm, t(9) = 2.24, p =
6/16/2019
HK Dr. Sasho MacKenzie

29. What if the t-test was two-tailed?
The null hypothesis would not be: HK students drink the same amount or less on the following Saturday, compared to the Saturday preceding a Biomechanics midterm.
But rather it would be: HK students drink the same amount on the following Saturday, compared to the Saturday preceding a Biomechanics midterm.
6/16/2019
HK Dr. Sasho MacKenzie

30. Graphic of two-tailed dependent sample t-test
t distribution for N = 10
From TDIST, p = .0516
Whoa!
We no longer have significance at  =.05
Combined area beyond t=2.24 and t = is .0516 -4 -3 -2 -1 1 2 3 4 t
6/16/2019
HK Dr. Sasho MacKenzie

31. Interpreting the P-value
In an experiment of this size, if the populations really have the same mean, what is the probability of observing at least as large a difference between sample means as was, in fact, observed?
There is a p% chance of observing a difference as large as you observed even if the two population means are identical (the null hypothesis is true).
Random sampling from identical populations would lead to a difference smaller than you observed in 1-p% of experiments, and larger than you observed in p% of experiments.
6/16/2019
HK Dr. Sasho MacKenzie