INTRODUCTION TO T-TESTS
Outline
INTRODUCTION TO THE T-DISTRIBUTION
A. WHAT HAVE WE DONE SO FAR?
B. t-test for one sample mean
C. Sampling distribution of t
D. Step-by-Step example of t-test
E. Additional considerations
F. t-test for two independent sample means
G. Step-by-Step example of 2 sample t-test
Introduction to The t-distribution
A. What we have done so far:
• Compared one sample mean to known population (
• Assumed (2 (and ([pic]) were known exactly
• In reality, (2 is rarely known!
• Must use s2, our unbiased estimate of (2
• Can’t use z-statistic, need to use t-statistic
Research Problem:
Does time pass more quickly when we’re having fun?
Perform a pleasant task for 12 minutes.
Judge how much time has passed.
Will judgement differ significantly from 12 on average?
B. t-test for One Sample Mean
t = [pic]
where: [pic] or [pic]
[pic] replacement for [pic]
• t is a substitute for z whenever ( is unknown
• s2 estimate for (2 in the formula for standard error
• s[pic] serves as estimate of ([pic]
• df = n-1
C. Sampling Distribution of t
• a family of distributions, one for every (n)
• Not actually n, but degrees of freedom
df = n –1
larger df = t gets closer to normal curve
larger df = s2 better estimate for (2
• A sampling distribution of a statistic
• All possible random samples of size n
• symmetric, unimodal, bell-shaped
• ( = 0
• Major difference between t and z:
tails of t are more plump
center is more flat
Table of the t-distribution:
Table B.2
• Values represent critical values (CV)
• Values mark proportions in the tails
• t is symmetric, so negative values not tabled
Need three things:
(1) one-tailed or two-tailed hypothesis
(2) alpha (α) level of significance
(3) degrees of freedom (df)
• Caution, not all values of df in table
If df not there, use CV for smaller df
example: df = 43, use CV for df = 40
df = 49, use CV for df = 40
Step-by-Step Example
(1) Does a pleasant task affect judgments of time passed?
Hypothesized: (hypoth = 12
Sample data: [pic] = 10.5 s2 = 2.25 n =10
(2) Statistical Hypotheses:
assume two-tailed
H0:
H1:
(3) Decision Rule:
α =
df = (n-1)
Critical value from Table B.2 =
(4) Compute observed t for one sample mean:
t = [pic] where [pic]
s[pic] = t =
(5) Make a decision:
(6) Interpret results:
E. Additional Considerations
Bottom Line:
• Use t whenever ( is unknown
• one-sample t when you have one sample mean compared against an hypothesized population (
• Your (hyp can be any reasonable value to test against
• Hypothesis testing steps with t are same as with z
• Differences:
• use s2 to estimate (2
• need df (n-1) to determine critical value
What Does All this Mean in Practical Terms?
• Critical values for t will be larger than for z
example: α = .05, two-tailed
zcrit = ( 1.96
tcrit = ( 3.18 when df = 3
tcrit = ( 2.04 when df = 30
tcrit = ( 1.98 when df = 120
• Larger n, less discrepancy between t and z
• Larger n, smaller critical value
• Larger n, more power! Easier to reject H0
Reporting t in the literature:
t (9) = 3.19, p < .05
Written scientific report:
“When distracted by a pleasant task, participants significantly underestimated how many minutes had passed (M = 10.5 minutes, SD = 1.5) during the 12 minute task period, t(9) = -3.16, p < .05, two-tailed.”
• Reject H0 ( p < .05 “statistically significant”
Retain H0 ( p > .05 “not statistically significant”
More about variance:
• For inferential statistics, variance is bad
• Big differences between scores, makes it hard to see trends or patterns
• High variance = noise, confusion in the data
• High variance = larger standard error
• When variance large, more difficult to obtain a statistically significant outcome. Effect size estimates will be smaller as well.
• What can we do?
Decrease experimental error/noise
Increase sample size to reduce standard error
More about p-values:
t (9) = 3.19, p < .05
• p = probability of making Type 1 Error
p = probability in the tail
> or < your alpha level
• Reject H0 ( p < α
Retain H0( p > α
• p < α observed t falls in critial region (in the tails)
• p > α observed t does not fall in critial region
t-test for Difference Between Two
Independent Sample Means
Typically have two sample means:
example:
Does new drug reduce depression?
Placebo: [pic]1 = 30
New Drug: [pic]2 = 25
Compare two sample means…
are they from the same population….
are the differences simply due to chance?
Research Problem:
Are people less willing to help when the person in
need is responsible for his/her own misfortune?
“Please take a moment to imagine that you're sitting in class one day and the guy sitting next to you mentions that he skipped class last week to go surfing (or because he had a terrible case of the flu). He then asks if he can borrow your lecture notes for the week. How likely are you to lend him your notes?”
1-------2-------3-------4-------5-------6-------7
I definitely would I definitely WOULD
NOT lend him my notes lend him my notes
High responsibility ( “went surfing”
Low responsibility ( “had a terrible case of the flu”
UCSB class data:
High responsibility: [pic]1 = 4.65 s2 = 2.99
Low responsibility: [pic]2 = 5.34 s2 = 2.06
Hypotheses Testing with Two Sample t
A. Statistical Hypotheses (two-tailed):
H0: (1 = (2
H1: (1 ( (2
Alternative form: H0: (1 - (2 = 0
H1: (1 - (2 ( 0
One-Tailed Hypotheses
Upper tail critical:
H0: (1 ( (2
H1: (1 > (2
Lower tail critical:
H0: (1 ( (2
H1: (1 < (2
Logic of the new t-test:
t = [pic]
[pic]1 approximates µ1 with some error
[pic]2 approximates µ2 with some error
Error for one sample mean: [pic]= [pic]
Error for two sample means: [pic]= [pic]
Note use of equivalent symbols:
[pic] = sdiff = s (M1-M2)
B. Computing independent measures t-statistic:
t = sample mean difference
estimated standard error
observed t statistic:
t = [pic]
degrees of freedom:
df = (n1 - 1) + (n2 – 1)
estimated standard error (when n1=n2):
[pic]= [pic]
What if sample sizes are unequal?
Take weighted average of the two variances
(a pooled variance estimate):
Step 1: pooled variance estimate:
[pic]
or [pic]
Step 2: standard error:
[pic]= [pic]
T-Test for Two Independent Sample Means
SAMPLE DATA:
High Resp: [pic]1 = 4.65 Low Resp: [pic]2 = 5.34
(Surfing) n1 = 74 (Flu) n2 = 73
s12 = 2.99 s22 = 2.06
(1) Research Question/Hypothesis:
IV =
DV =
(2) State the Statistical Hypotheses:
H0:
H1:
(3) Create a Decision Rule:
(a) ( = .05
(b) two-tailed test
(c) df = (n1 – 1) + (n2 – 1)
Closest df = , so critical value is
(4) Compute the Observed t:
(a) First compute pooled variance:
[pic] or [pic]
(b) Compute standard error: sdiff = [pic]
sdiff =
(c) Compute t: t = [pic] =
(5) Decision:
(6) Interpretation:
Homework Problems
CHAPTER 9: 7, 8, 12
CHAPTER 10: 8 (NOT SECTION E), 10, 13
-----------------------
DF
observed
t-value
α probability in the tail
?
t
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Appendix B.2: The t Distribution
Table 9-1 (p. 285). A portion of the t-distribution table. The numbers in the table are the values of t that separate the tail from the main body of the distribution. Proportions for one or two tails are listed at the top of the table, and df values for t are listed in the first column.
Critical values of t for a two-tailed test, df =3, ( = .05
+3.182
(3.182
.025
.025
?
t
[pic]
[pic]
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