1. Quantitative Methods in HPELS HPELS 6210
Single-Sample T-Test
Quantitative Methods in HPELS
HPELS 6210

2. Agenda Introduction The t Statistic
Hypothesis Tests with Single-Sample t Test
Instat
Assumptions

3. Introduction Recall  Inferential statistics:
Use the sample to approximate population
Answer probability questions about H0
Z-score one example of an inferential statistic
Must have information about the population standard deviation!

4. Introduction The Problem with Z-Scores:
In most cases, the population standard deviation is unknown
In these cases, an alternative statistic is required in order to test a hypothesis

5. Agenda Introduction The t Statistic
Hypothesis Tests with Single-Sample t Test
Instat
Assumptions

6. The t Statistic Estimation of the Standard Error (M): Recall:
Population SD is unknown  SEM must be estimated with information from the sample only
Recall:
Sample variance = s2 = SS / n-1 = SS / df
Sample SD = s = √SS / n-1 = √SS / df
Therefore:
Estimated SEM = sM = s / √n = √s2 / n

7. The t Statistic Calculation of the single-sample t-test:
Formula similar  Z-score however
SEM (M)  estimated SEM (sM):
t = M - µ / sM
t = M - µ / √s2 / n
Degrees of Freedom:
Similar to Z-score, only n-1 values are free to vary
As sample size increases:
Estimated SEM (sM)  more accurate representation of SEM (M)
t statistic  more accurate representation of Z

8. The t Distribution Recall  Z distribution
With infinite samples, the sampling distribution:
Approaches normal distribution
µ = µM
This is also true for the t distribution.

9. The t Distribution The shape of the t distribution:
Changes as df changes
A “family” of t distributions exists
Distribution more normal as df increases
(Figure 9.1, p 284)
Characteristics of a t distribution:
Symmetrical and bell-shaped
µ = 0

10. Normal distribution has less variability than the t distribution
Why?

11. The t Distribution
Z distribution  SEM is calculated and is therefore constant
t distribution  SEM is estimated and is therefore variable
As df increases:
Estimated SEM (sM) resembles SEM (M)

12. Agenda Introduction The t Statistic
Hypothesis Tests with Single-Sample t Test
Instat
Assumptions

13. Hypothesis Test: Single-Sample t-Test
Example 9.1 (p 288)
Overview:
Direct eye contact is avoided by many animals
Moths have developed large eye-spot patterns to ward off predators
Researchers want to test the effect of exposure to eye-spot patterns on the behavior of moth-eating birds
Birds are put in a room (60-min) with two chambers, separated by a doorway (Figure 9.4, p 289)
If no effect  equal time in each chamber (Figure 9.3, p 287)

14. Hypothesis Test: Single-Sample t-Test
Recall  General Process:
State hypotheses
Set criteria for decision making
Sample data and calculate statistic
Make decision

15. Assume:
n = 16
M = 39 minutes
SS = 540
 = ?  use the t-test
Step 1: State Hypotheses
H0: µplainside = 30 minutes
H1: µplainside ≠ 30 minutes
Step 2: Set Criteria for Decision
Alpha (a) = 0.05
Critical value?

16. 1st Column: df = n – 1
1st Row: Proportion located in either tail
2nd Row: Proportion located in both tails
Body: The critical t-values specific to df and alpha
1st Column: df = 16 – 1 = 15
1st Row: Ignore
2nd Row: 0.05 (alpha)
Body: ?

17. 0.05 15
2.131

18. What would the distribution look like if df were larger?
Step 3: Calculate Statistic
Variance (s2)
s2 = SS/df
s2 = 540/15
s2 = 36
Step 3: Calculate Statistic
SEM (sM)
sM = √s2 / n
sM = √36 / 16 = √2.25
sM = 1.50
Step 3: Calculate Statistic
t statistic
t = M - µ / sM
t = 39 – 30 / 1.5 = 9 / 1.5
t = 6.0
Step 4: Make a Decision
t = 6.0 >  Accept or Reject?

19. Confirmation of decision

20. One-Tailed Single-Sample t-Test Example
Example 9.4 (p 297)
Overview:
Researchers are still interested in the effect of eye-spot patterns on bird behavior
Based on prior knowledge researchers assume birds will spend less time with eye-spot patterns
Therefore a directional (one-tailed test) will be used

21. Step 2: Set Criteria for Decision
Alpha (a) = 0.05
Critical value?

22. 0.05 15
1.753

23. Step 3: Calculate Statistic
Same as last example
t = 6.0
Step 4: Make Decision
t = 6.0 > 1.753
Accept or Reject?

24. Agenda Introduction The t Statistic
Hypothesis Tests with Single-Sample t Test
Instat
Assumptions

25. Instat Type data from sample into a column. Choose “Statistics”
Label column appropriately.
Choose “Manage”
Choose “Column Properties”
Choose “Name”
Choose “Statistics”
Choose “Simple Models”
Choose “Normal, One Sample”
Layout Menu:
Choose “Single Data Column”

26. Instat Data Column Menu: Parameter Menu: Confidence Level:
Choose variable of interest
Parameter Menu:
Choose “Mean (t-interval)”
Confidence Level:
90% = alpha 0.10
95% = alpha 0.05

27. Instat Check “Significance Test” box:
Check “Two-Sided” if using non-directional hypothesis.
Enter value from null hypothesis.
What population value are you basing your sample comparison?
Click OK.
Interpret the p-value!!!

28. Reporting t-Test Results
How to report the results of a t-test:
Information to include:
Value of the t statistic
Degrees of freedom (n – 1)
p-value
Example:
The average IQ of Black Hawk County 6th graders was significantly greater than 75 (t(100) = 2.55, p = 0.02)

29. Agenda Introduction The t Statistic
Hypothesis Tests with Single-Sample t Test
Instat
Assumptions

30. Assumptions of Single-Sample t-Test
Independent Observations:
Random selection
Normal Distribution:
Tenable if the population is normal
If unsure about population  assume normality if sample is large (n > 30)
If the sample is small and unsure about population  assume normality if the sample is normal
Tests are also available
Scale of Measurement
Interval or ratio

31. Violation of Assumptions
Nonparametric Version  Chi-Square Goodness of Fit Test (Chapter 17)
When to use the Chi-Square Goodness of Fit Test:
Scale of measurement assumption violation:
Nominal or ordinal data
Normality assumption violation:
Regardless of scale of measurement

32. Textbook Assignment
Problems: 3, 11, 23, 27