1. Topics
Today: Case I: t-test single mean: Does a particular sample belong to a hypothesized population?
Thursday: Case II: t-test independent means: Are two sample means drawn from an identical population or different populations?

2. Z-Test vs t-Test Z-Test T-Test
Where population standard deviation and standard error are known
Where sample size is > 30
Where the normal curve is the model for the sampling distribution for determining the probability of obtaining our result under the null hypothesis
T-Test
Where population standard deviation and standard error are not known and have to be estimated from the sample data
Where the t distribution is model for sampling distribution for determining probability of our results under the null hypothesis

3. Z and t Compared

4. Z and t Sampling Distributions

5. Degrees of Freedom (dfs)
t distributions differ according to their degrees of freedom (based on the sample size)
For single sample case the t distribution is based on n-1 dfs
For 2 sample case the t distribution for differences is based on N-2 dfs (i.e., n -1 for each sample)

6. Hypothesis Testing Using t-Tests
Identify population of interest
Draw samples (preferably probability samples)
Set up null and alternative hypotheses
Select level of significance (e.g., .05, .01)
Calculate sample statistic (e.g. mean)
Calculate standard error of the sample statistic
Convert observed mean to standard error points
Determine t-critical value (based on level of significance chosen)
Compare t-observed value against the t-critical value
Decide: “Reject” or “Do not reject” null hypothesis

7. Sampling Distribution of Means: Standard Errors, Critical Values, and Ps
t Distribution
Sample Size =31
df=30
 = .05 or .01
Two tailed
Test 
-2se
-1se
+1se
+2se
Critical
Values
-2.75se
-2.04se
+2.04se
+2.75se
p =
< = .05 = outside 2.04 on either end
< = .01 = outside of 2.57 on either end

8. Sampling Distribution of Means: Standard Errors, Critical Values, and Ps
t Distribution
Sample Size =31
df=30
 = .05 or .01
One tailed
test u
-2se
-1se
+1se
+2se
Critical
Values se
+2.457se
p =
< = .05
< = .01

9. Sampling Distribution of Means: Standard Errors, Critical Values, and Ps
t Distribution
Sample Size =31
df=30
 = .05 or .01
One tailed
test u
-2se
-1se
+1se
+2se
Critical
Values
-2.457se se
p =
< = .01
< = .05

10. Case I: t-Test
Does a particular sample belong to a hypothesized population?
Draw single sample from population
Calculate sample statistic such as mean
Test null hypothesis of no difference between sample and population mean

11. Case I: Assumptions Scores randomly sampled from some population
Scores in the population are normally distributed

12. Example t-Test Single Sample: SAT Data: UCLA from OAS

13. t-Test for Single Sample Designs
Set Null Hypothesis:
Set Alternative Hypothesis:
Decide Significance Level: .01
Compute Standard Error:
Compute tobserved
Locate tcritical with N-1df:
Decide
Reject H0 if tobserved >= tcritical :
Do Not Reject H0 if tobserved < tcritical :
Conclude:

14. UCLA: Sampling Distribution Picture
t Distribution
Sample Size =10
df=9
 = .01
One tailed
test
3.95 
-2se
-1se
+1se
+2se
= 1000
Critical
Values
+2.82se
p =
= < .01

15. X - tcritical(.01/1,9) (sx) <=  <= X+ tcritical(.01/1,9) (sx)
99% Confidence Interval
X - tcritical(.01/1,9) (sx) <=  <= X+ tcritical(.01/1,9) (sx)
(21.26) <=  <= (21.26)
<=  <=
<=  <=
Can feel 99% confident that this interval includes the population mean for the UCLA undergraduates. Since it does not include the “known” population mean of 1000 we can conclude that UCLA undergraduates have on average higher total SAT scores than the population of high school graduates taking the SAT.