Download T-Tests for Case I and Case II Research

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?
Z-Test vs t-Test
• Z-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
Z and t Compared
So m eCo m p ariso ns
Z
t
(Pop u at
l ion s d known) (Pop u at
l ion s d unk n own)
S tandard Error
Tes t S tat i s tci
Observed
Cri t cal
i Values
Z val ues under normal
curve (Z Tab le)
t values under curves of r
di fferent sample sizes (t
Tab le)
Z and t Sampling Distributions
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)
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
Sampling Distribution of Means: Standard
Errors, Critical Values, and Ps
t Distribution
Sample Size =31
df=30
 = .05 or .01
-2se
Critical
Values
p=
-2.75se
Two tailed
Test
-1se
-2.04se
< = .05 = outside 2.04 on either end

+1se
+2se
+2.04se +2.75se
< = .01 = outside of 2.57 on either end
Sampling Distribution of Means: Standard
Errors, Critical Values, and Ps
t Distribution
Sample Size =31
df=30
 = .05 or .01
-2se
One tailed
test
-1se
u
+1se
+2se
Critical
Values
+1..697se
+2.457se
p=
< = .05
< = .01
Sampling Distribution of Means: Standard
Errors, Critical Values, and Ps
t Distribution
Sample Size =31
df=30
 = .05 or .01
-2se
Critical
Values
p=
-2.457se
< = .01
One tailed
test
-1se
-1..697se
< = .05
u
+1se
+2se
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
Case I: Assumptions
• Scores randomly sampled from some
population
• Scores in the population are normally
distributed
Example t-Test Single Sample: SAT
Data: UCLA from OAS
Stud ent
Scor e
Stud ent
Scor e
1
1050
6
1075
2
1200
7
1100
3
1100
8
1025
4
1130
9
1000
5
1160
10
1000
X=1084
S=67.24
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:
UCLA: Sampling Distribution Picture
t Distribution
Sample Size =10
df=9
 = .01
One tailed
test
3.95
-2se
Critical
Values
p=
-1se

= 1000
+1se
+2se
+2.82se
= < .01
99% Confidence Interval
X - tcritical(.01/1,9) (sx) <=  <= X+ tcritical(.01/1,9) (sx)
1084 - 2.82 (21.26) <=  <= 1084 + 2.82 (21.26)
1084 - 59.95 <=  <= 1084 + 59.95
1024.1 <=  <= 1149.95
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.