1. t-Tests
Chapters 14 and 13

2. Quiz
You measure the stress level of 7 employees in a division undergoing major restructuring to see if their stress level is different than the stress level of the average employee. Their average stress level was 7.6, higher than the 5.5 average for the average employee.
Your null hypothesis is that
Stress of division members = stress of average employee.
You do a z-test and a get a z of 4. Your zcrit is 1.96?
Do you reject the null hypothesis?

3. rcrit: Testing the significance of r
For 27 people in a statistics class, the correlation between shoe size and extraversion is
Can we be 95% sure there is really a relationship between shoe size and extraversion?
That means that r ≠ 0 (2 tail hypothesis).
Look up rcrit in Table E “Critical Values of Pearson r.”
Degrees of freedom (df) = N - # of distributions = N – 2
df = 27 – 2 = 25
α = .052 tails => rcrit =
|r| < rcrit, so we retain the null hypothesis. The correlation is perhaps spurious.

4. Table of Critical Values of r

5. Student Correlations. . .
A. Among 62 stats students, the correlation between Christian Commitment and Extraversion is
Is this a significant correlation (alpha = .05, 2 tails)?
B. Among this same group, the correlation between Gender (0 = male, 1 = female) and Shoe Size is

6. Excel: Testing the significance of r
To test the significance of the correlation between two variables (e.g., Justin Bieber and age), use Regression in Data Analysis to obtain both the correlation and the p value (2 tails).
See File Chapter 13 Correlation Data.xlsx.

7. Reporting Correlations in APA Format
Hypothesis: “There will be a correlation between liking Justin Bieber and age.”
Results: “The Pearson correlation between liking Justin Bieber and age (N = 25) was significant, r(23) = -.46, p = .02.”
Note: If we had made a one tail hypothesis (“The correlation between liking Justin Bieber and age will be negative”), we would need to divide the p value by 2.

8. Testing the significance of r: Online Calculators
Google something like “Online significance r calculator”
Enter N and r
Example:

9. Class Exercise
Test the following hypotheses using Chapter 12 Correlation Data.xlsx
There is a relationship between age and liking Star Wars.
There is a relationship between gender and liking the Kardashians, specifically, females like the Kardashians more than males.

10. z-Test vs. t-Test
In a z-test, we compared a sample mean to the population mean. We needed to know the population standard deviation.
But we rarely have the population standard deviation.
In a t-test, we compare a sample mean to some other mean (such as a different sample or the population) when we don’t know the population standard deviation.
Much more versatile.
For example, we can tell if there is a difference between two groups: Is the division led by Al more or less stressed than the division led by Bob?

11. z-Test vs. t-Test
When we don’t know the population standard deviation, the sampling distribution is not quite normally shaped.
It’s described
by the t-statistic,
not the z-statistic.
Its shape depends on the degrees of freedom (df), which is a function of the sample size.

12. Type 1 (Chapter 14): Two-Sample (independent) t-tests.
Suppose Al’s division has higher turnover than Bob’s division. You want to find out if people in Al’s division are more stressed out than people in Bob’s.
Hypothesis: Stress of Division A (Al) is higher than the stress of Division B (Bob’s).
This is a one tailed hypothesis.
Null hypothesis: The stress of Division A is the same or lower than the stress of Division B.

13. Type 1 (Chapter 14): Two-Sample (independent) t-tests.
The Data: Stress of Two Groups.xlsx
Calculate descriptive statistics for it.

14. Type 1 (Chapter 14): Two-Sample (independent) t-tests.
To perform the t-test, use Data Analysis, t-Test: Two Sample Assuming Equal Variances.
Set “Hypothesized Mean Difference” to 0.
Reporting and interpreting the results. . .

15. APA formatted results of two sample t-test.
The stress level of Division A (M = 13.75, SD = 4.06, n = 8) was hypothesized to be greater than the stress level of Division B (M = 9.63, SD = 3.74, n = 8). This difference was significant, t(14) = 2.14, p = .03 (1 tail).

16. Equation for t for two independent samples (groups)
For testing if there is a difference between the variable of interest in two groups

17. t–tests for two independent samples (groups)
Example: Are bears cuter than cats?
Hypothesis: The average
cuteness of bears is greater than the average cuteness of cats.

18. Separating Data into 2 Columns
Often, data for two groups is collected in a single column, with another column indicating the grouping.
Do men like Star Wars more than women?
One tailed hypothesis: Men like Star Wars more than Women.
Chapter Star Wars and Gender.xlsx
Use the filtering option in the table.
Copy relevant info below the table
(to not miss lines).

19. Special Cases of t-Tests (Chpt 13)
Special case number 1: Paired two-sample t-Tests
When you have two data points for each unit of analysis.
Examples:
OCBs (Organizational Citizenship Behaviors) before and after a move to a new office.
Marital satisfaction scores of husbands and wives in a specific church.
Organizational justice (perceptions of how fair an organization is) before and after a major change.

20. Paired Sample t test
Example: OCB ratings made by a supervisor before and after a move to a new location.
“How characteristic is it of this employee to help others who are in need?”
Chapter Paired Samples t-Test OCBs.xlsx
Use Data Analysis, “T-test:
Paired Two-Sample for Means.”
Directional hypothesis: OCBs
will decrease after the move.

21. Paired Sample t test Results
OCBs were hypothesized to decrease after a change of office location. In a small sample (N = 5), OCBs decreased after a move from M = 7.00, SD = 1.58 to M = 5.6, SD = However, this change was not significant, t = 1.09, p = .15, one tail.
Note: this t-test can also be called “matched pair” or “repeated measures.”

22. Special Cases of t-Tests
Special Case No. 2: Single Sample t-tests.
These are like z-tests when we don’t know the population SD
We will compare the sample mean to the population mean.

23. Sunday School example
During the last school year (Sept-June), the average Sunday School Attendance for kids K-5 was 20.
In August, your church has a Vacation Bible School with the goal of reaching neighborhood kids.
In September, the attendance for the first 4 Sundays is: 28 18 26 24
Can you be 90% sure the Sunday School grew? α = .10

24. Sunday School Example Use Data Analysis, Paired Two-Sample t-Test.
Directional hypothesis: The Sunday School has grown.
Assign the population mean to the second score of each data point (treating it like a homogenous 2nd sample).
Data: Chapter Sunday School Example.xlsx

25. A z-test or a t-test? Test Population Mean μ
Population Standard Deviation σ
Sample Mean or
Sample Standard Deviation s or s1
2nd Sample Mean
2nd Sample Standard Deviation s2
Formula
N = sample size when there’s just one sample.
n1 and n2 are the sample sizes when there are 2 samples.
z-test
Chpt 12 √
Single sample
t-test
Chpt 13
Two sample (Ind. samples) t-test
Chpt 14

26. Confidence Intervals
A Confidence Interval is a range of values that probably contains the true value.
Confidence limits are the values that bound the confidence interval.
The 95% Confidence Interval of the mean is an interval such that the probability is 95% that the interval contains the true population mean.

27. Confidence Interval of the Mean
How much do students really like Justin Bieber and the Kardashians.
Use Data Analysis, Descriptives to get the 95% confidence interval (95% CI).
Check the boxes for both “Summary statistics” and “Confidence Level for Mean”
Chapter Kardashians and Bieber.xlsx

28. Confidence Interval of the Mean
95% CI for Justin Bieber = /- .59
95% CI [1.93 – 3.11]
95% CI for the Kardashians = /- .40
95% CI [1.02 – 1.92]
Interpretation: We can be 95% confident that the true population mean of Bieber is between 1.93 and 3.11
No overlap roughly means that the difference is statistically significant.

29. Confidence Intervals for Proportions: Using an Online Calculator
Google the test you want to use: “confidence interval for proportions calculator”
Use a calculator that you can figure out.