T-Test Chp 9 and 10 (Spg 2005) 1
t
The kind of research tt we’ll see the most of now is where we don’t know anything abt the pop(s) that sample(s) come from.
3 Different t -test scenarios
1. compare two samples to each other (Independent samples t-test —chp 10)
2. compare pretest scores to post-test scores with one group (Dependent t-test)
3. compare a sample to a pop (one-sample t-test / t-test for a single sample)
When do we use a t-test?
When we don’t know the variance (SD squared) of the population.
If we know the variance of the pop, then we use a z -test
1: Restate the Question as a Research Hypothesis / Alternative Hypoth (H
1
) and a Null Hypoth (H o
) about the Population
2: Determine the Characteristics of the Comparison Distribution
*We have to estimate the pop variance b/c it’s unknown.
*We will have to use a t-distribution b/c the shape of the distribution is not assumed to be normal like it was with the z dist. (Look at Fig 9-2 on page. 305)
When we estimate pop variance with a sample, we have the potential for error.
The mathematical effect is tt there’s likely to be slightly more extreme means than in a normal distribution of sample means.
The smaller the sample size the potential for more error b/c we have less info than we’d have w/ a larger sample.
“30” seems to be the magic sample size cutoff #.
The variance of samples are usually smaller than the variance in a pop, so if we want to increase the likelihood tt our sample is the same as the pop (null hypoth) then an adjustment needs to be made.
We divide the sum of squared deviations by the sample size minus 1.
Degrees of freedom (df): number of scores free to vary when estimating a pop parameter; usually part of a formula for making that estimate.
EX: In the formula for estimating the pop variance from a single sample, the df is the number of scores minus 1.

T-Test Chp 9 and 10 (Spg 2005) 2
3: Determine the Cutoff Sample Score (Critical Value) on the Comparison
Distribution (Sampling Dist) at Which the Null Hypoth Should Be Rejected
B/c the dist is not assumed to be normal we will use a new table to find the cutoff sample score (critical value) t cv
.
The t dist is actually a “family of distributions” where a separate dist exists for each df. When using a t table, we’ll need to know: a) is the hypoth test one-tailed or two tailed? b) what is the sig level? .01? .05? c) how many df are there? sample size minus 1.
4. Determine Your Sample’s Score on the Comparison Dist t = (sample mean
– pop mean) / stand dev of sample mean (SE) t

X
  s
X
SE = s/sqr root of n s s
X
= n
5: Decide Whether to Reject the Null Hypoth or Retain the Null Hypoth
EX: Self Monitoring Data (SPSS Raw Data provided by BJ)
TAs and Absent students: We worked out an example in class where the students were told that the pop mean (all UNC students) = 6.
The students actually obtained the following descriptive stats: sample size: sample mean: sample stan dev: standard error:

T-Test Chp 9 and 10 (Spg 2005) 3
T
1: Restate the Question as a Research Hypothesis / Alternative Hypoth (H
1
) and a Null Hypoth (H o
) about the Population
Pop 1:
Pop 2:
Hypotheses written out:
H o
:
H
1
:
Hypotheses using statistical notation
For a nondirectional one sample t test:
H o
: μ
1
= μ
2
H
1
: μ
1
≠ μ
2
2: Determine the Characteristics of the Comparison Distribution
You will be given the pop mean, but you may or may not be given the pop SD
(variance information).
If you do know the pop SD (or variance), then you use a Z distribution.
If you do NOT know the pop SD (or variance), then you use a t distribution.
You then have to estimate the pop SD from what you know of your sample's SD.
μ =
(Here mu refers to the pop mean) s = (Here s is the sample standard deviation) s
X
= (Here s
X
refers to the standard error of the mean; it also refers to the standard deviation of the distribution of means; it is abbreviated SE in journal articles) s s
X
= n

T-Test Chp 9 and 10 (Spg 2005) 4
3: Determine the Cutoff Sample Score (Critical Value) on the Comparison
Distribution (Sampling Dist) at Which the Null Hypoth Should Be Rejected.
What three pieces of information (order doesn't matter) do you need to know to complete this step?
1. Degrees of Freedom (n – 1):
2. Directionality of hypothesis (one-tailed or two-tailed; recall that two-tailed is more conservative):
3. Significance Level (alpha; α) ; (typically we set our alpha level at .05 or .01):
The information obtained from this step is what will allow you to shade the rejection region(s).
If the hypothesis test is two-tailed, your t cv
will be + . t cv
= the critical value(s) of t . You will plot this number(s) on the test statistic distribution (-2, -1, 0, 1, 2).
4. Determine Your Sample’s Score on the Comparison Dist
Recall the comparison distribution is a type of distribution used in hypothesis testing. It represents the pop situation if the null hypothesis is true. It's the distribution to which you compare the score based on your sample's results.
When you sketch the comparison distribution, you need to place the pop mean in the middle of your distribution. You will demarcate* this distribution by using the standard error ( s
X
).
*Demarcate: to set or mark the limits of; delimit; to mark the difference btwn; distinguish; separate. t obs

X
 s
X

X is your sample mean.
μ is your pop mean. s is the standard error based on your sample's SD and sample size.
X

T-Test Chp 9 and 10 (Spg 2005) 5 t obs
refers to the observed value of the t test statistic
t obs
= plot this number on the comparison distribution sketch on the t test number line (-2, -1, 0, 1, 2).
The t obs
should line up with the sample mean ( X ) on the null hypothesis distribution (where you placed the pop mean in the middle and demarcated the numberline by the SE).
5: Decide Whether to Reject the Null Hypoth or Retain the Null Hypoth
Does the observed value of t exceed the critical value(s) of t ? If so, then reject the null hypothesis and accept the alternative hypothesis.

T-Test Chp 9 and 10 (Spg 2005) 6
Example #1 (If TA, See attached materials provided for you)
High schools and state boards of education recently have put considerably greater emphasis on having students achieve a minimal level of competency before they can receive a diploma. In an attempt to defend the expenditure of money over the past year on tutorials and other forms of remedial educ, the superintendent of a large school district has collected scores from 20 of this year's seniors on a standardized competency exam.
Last year's mean score across the entire school system (the pop of interest) was 76.0. This year the mean ( X ) for the sample of 20 students was
79.3 with a standard deviation of 6.4. Can these data be taken as evidence tt the mean competency score improved, or does it represent a chance result to be commonly expected from a pop with μ = 76.0?
Despite the prediction that competency scores will improve, lets be conservative and state a nondirectional hypothesis. Lets set our significance level at .05. In other words, we're trying to make the probability of committing a
Type I error less than 5%.
H o
: μ = 76
H
1
: μ ≠ 76
1: Restate the Question as a Research Hypothesis / Alternative Hypoth (H
1
) and a Null Hypoth (H o
) about the Population
2: Determine the Characteristics of the Comparison Distribution
3: Determine the Cutoff Sample Score (Critical Value) on the Comparison
Distribution (Sampling Dist) at Which the Null Hypoth Should Be Rejected
4.
Determine Your Sample’s Score on the Comparison Dist
5: Decide Whether to Reject the Null Hypoth or Retain the Null Hypoth

T-Test Chp 9 and 10 (Spg 2005) 7
What are degrees of freedom?
Df are related to the number of scores, items, or other units in a data set and the idea is that these scores, etc. are “free to vary”
A participant could get a 0 or a 1 or a 2 or 18 or 24 or 25, etc.
If you have 40 scores you have 40 degrees of freedom b/c each of the 40 participants’ scores are free to vary.
That is, before collecting data we don’t know what any of these scores will be.
Once we calculate the mean, we lose one degree of freedom.
For example: Once we know the mean and all except one of the scores, we can figure out that last score, so it’s not free to vary.
When calculating the sd or variance, you are calculating how much the scores vary around the sample mean.
B/c the sample mean is known, one df is lost, and the dfs become n-1, the number of items in the set less one.
In a sample size of 40, where you have 40 scores (for each participant you have a corresponding score) and you know the mean, so one score isn’t free to vary any more b/c we can figure it out. So we’re left with 39 df b/c these 39 scores are
"free to vary".
Additional info on df:
Consider a rsrch example w/ 5 scores whose mean is 10.
Four of these scores can be anything you want
(18, 18, 16, 2), but the 5 th score cannot be chosen freely. It must be -4 if the mean is going to be 10.
In other words there are only four free numbers in tt set of five scores, once the mean is determined. Therefore, we have four degrees of freedom. N
– 1 = 5 – 1
= 4
INFORMATION REPEATED BRIEFLY
The Student’s t Distribution: General Information

hx (Guinness beer)
 accounts for the dist of the sampling means not being normal. So we can’t use the z dist.

There is only one z dist, but there are a family of t distributions depending on the sample size.

Finding the critical value for our statistical purposes will depend on knowing the sample size.

T-Test Chp 9 and 10 (Spg 2005) 8

We want the sample size to be a good size. Over 30 is good, but bigger sample sizes are better.

As the sample size gets bigger, the t dist approaches normality like the z dist.

Degrees of freedom are related to sample size.
One sample t test Brief Info
Mostly we rarely know the value of "σ" and usually will have to estimate it by way of the sample standard deviation "s"
When we replace with s in the formula the nature of the test changes.
We can no longer declare the answer to be a z score and evaluate it with reference to tables of z.
Instead we'll denote the answer as t and evaluate it with respect to tables of t, wch are somewhat different.
B/c the t test uses s 2 as an estimate of σ 2 , it's important tt we first look at the sampling dist of s 2 . t

X
  s
X
Recall that s
X
 s
N
 s
2
N

T-Test Chp 9 and 10 (Spg 2005) 9
Def: A hypothesis testing procedure in which each subject is measured twice.
In other words, the mean for each group of scores (of which there are 2 sets) are dependent on each other b/c they're from the same people.
Aliases for Dependent Sample T -Test

T Test for Dependent Means

T Test for Paired Samples

T Test for Correlated Means

T Test for Matched Samples
What do you differently for this type of T -Test compared to the One Sample T -
Test?
Two things:
1) You're going to use something called "Difference Scores."
2) You're going to assume the pop mean is 0.
T
1: Restate the Question as a Research Hypothesis / Alternative Hypoth (H
1
) and a Null Hypoth (H o
) about the Population
Pop 1:
Pop 2:
Hypotheses written out:
H o
:
H
1
:
Hypotheses using statistical notation
For directional and nondirectional dependent sample t tests:
H o
: μ
1
-
μ
2
= 0
H
1
: μ
1
- μ
2
≠ 0

T-Test Chp 9 and 10 (Spg 2005) 10
2: Determine the Characteristics of the Comparison Distribution
You will be given the pop mean, but you may or may not be given the pop SD
(variance information).
If you do know the pop SD (or variance), then you use a Z distribution.
If you do NOT know the pop SD (or variance), then you use a t distribution.
You then have to estimate the pop SD from what you know of your sample's SD.
μ =
0 (Here mu refers to the pop mean) s
D
= (Here s is the sample standard deviation of the Difference) s
D
= (Here s
X
refers to the standard error of the mean; it also refers to the standard deviation of the distribution of means; it is abbreviated SE in journal articles) s s
D
= n
3: Determine the Cutoff Sample Score (Critical Value) on the Comparison
Distribution (Sampling Dist) at Which the Null Hypoth Should Be Rejected.
What three pieces of information (order doesn't matter) do you need to know to complete this step?
1. Degrees of Freedom (n – 1):
2. Directionality of hypothesis (one-tailed or two-tailed; recall that two-tailed is more conservative):
3. Significance Level (alpha; α) ; (typically we set our alpha level at .05 or .01):
The information obtained from this step is what will allow you to shade the rejection region(s).
If the hypothesis test is two-tailed, your t cv
will be + . t cv
= the critical value(s) of t . You will plot this number(s) on the test statistic distribution (-2, -1, 0, 1, 2).

T-Test Chp 9 and 10 (Spg 2005) 11
4. Determine Your Sample’s Score on the Comparison Dist
Recall the comparison distribution is a type of distribution used in hypothesis testing. It represents the pop situation if the null hypothesis is true. It's the distribution to which you compare the score based on your sample's results.
When you sketch the comparison distribution, you need to place the pop mean in the middle of your distribution. You will demarcate* this distribution by using the standard error ( s
X
).
*Demarcate: to set or mark the limits of; delimit; to mark the difference btwn; distinguish; separate. t

D s
D
D is your sample mean.
μ = 0 so, it is not included in this formula like it was in the one sample t -test. s is the standard error based on your sample's SD and sample size.
D t obs
refers to the observed value of the t test statistic
t obs
= plot this number on the comparison distribution sketch on the t test number line (-2, -1, 0, 1, 2).
The t obs
should line up with the sample mean ( D ) on the null hypothesis distribution (where you placed the pop mean in the middle and demarcated the numberline by the SE).
5: Decide Whether to Reject the Null Hypoth or Retain the Null Hypoth
Does the observed value of t exceed the critical value(s) of t ? If so, then reject the null hypothesis and accept the alternative hypothesis.
T

T-Test Chp 9 and 10 (Spg 2005) 12
Example #2: Soup Company
A major soup manufacturing company wants to improve its image after a major celebrity sponsor was caught doing something inappropriate. The marketing dept develops a commercial and assesses a small sample of people's attitudes before and after viewing the commercial. They wanted to assess attititudes toward XYZ soup company using a [0 (neg) to 10 (pos)] attitude scale. α = .05; two-tailed.
1: Restate the Question as a Research Hypothesis / Alternative Hypoth (H
1
) and a Null Hypoth (H o
) about the Population
Pop 1: Ppl like those who are being examined in this study.
Pop 2: Ppl whose attitudes do not change from before to after watching the commercial.
Hypotheses written out:
H o
: There is no difference in ppl's atts toward XYZ soup company in comparing their before and after ratings of watching a commercial.
H
1
: There is a difference in ppl's atts toward XYZ soup company in comparing their before and after ratings of watching a commercial.
Hypotheses using statistical notation
For directional and nondirectional dependent sample t tests:
H o
: μ
1
-
μ
2
= 0
H
1
: μ
1
- μ
2
≠ 0
2: Determine the Characteristics of the Comparison Distribution
"It turns out that in most research you do not even know the pop's mean; plus, in most research situations you usually have not one set, but two sets, of scores.
These two things, not knowing the pop mean and having two sets of scores, almost always go together." p. 313 (Aron & Aron, 2003)
If you do NOT know the pop SD (or variance), then you use a t distribution.
You then have to estimate the pop SD from what you know of your sample's SD.
μ = 0 (Here mu refers to the pop mean) s
D
= 1.977 (Here s is the sample standard deviation of the Difference) s
D
= .57 (Here s
X
refers to the standard error of the mean; it also s refers to the standard deviation of the distribution of means; it is abbreviated SE in journal articles) s
D
= n
3: Determine the Cutoff Sample Score (Critical Value) on the Comparison
Distribution (Sampling Dist) at Which the Null Hypoth Should Be Rejected.

T-Test Chp 9 and 10 (Spg 2005) 13
What three pieces of information (order doesn't matter) do you need to know to complete this step?
1. Degrees of Freedom (n
– 1): 12 – 1 = 11
2. Directionality of hypothesis (one-tailed or two-tailed; recall that two-tailed is more conservative): two-tailed
3. Significance Level (alpha; α) ; (typically we set our alpha level at .05 or .01):
.05
The information obtained from this step is what will allow you to shade the rejection region(s).
If the hypothesis test is two-tailed, your t cv
will be + 2.201 . t cv
= the critical value(s) of t . You will plot this number(s) on the test statistic distribution (-2, -1, 0, 1, 2).
4. Determine Your Sample’s Score on the Comparison Dist
Recall the comparison distribution is a type of distribution used in hypothesis testing. It represents the pop situation if the null hypothesis is true. It's the distribution to which you compare the score based on your sample's results.
When you sketch the comparison distribution, you need to place the pop mean in the middle of your distribution. You will demarcate* this distribution by using the standard error ( s
X
).
*Demarcate: to set or mark the limits of; delimit; to mark the difference btwn; distinguish; separate. t

D s
D t

2 .
50

.
57
4 .
385
D is your sample mean.
μ = 0 so, it is not included in this formula like it was in the one sample t -test. s is the standard error based on your sample's SD and sample size.
D t obs
refers to the observed value of the t test statistic
t obs
= 4.385 plot this number on the comparison distribution sketch on the t test number line (-2, -1, 0, 1, 2).

T-Test Chp 9 and 10 (Spg 2005) 14
Note:
The t obs
should line up with the sample mean ( D ) on the null hypothesis distribution (where you placed the pop mean in the middle and demarcated the numberline by the SE).
5: Decide Whether to Reject the Null Hypoth or Retain the Null Hypoth
Does the observed value of t exceed the critical value(s) of t ? If so, then reject the null hypothesis and accept the alternative hypothesis.
Yes, the observed value of t ( t obs
= 4.385) exceeds the critical value of t ( t cv
= +
2.201), so we reject the null hypothesis and accept the alternative hypothesis.
Ppl's attitudes changed (became more positive) after watching the commercial.
In a journal article, this information would be written as:
Based on a t test for a dependent sample participants' attitudes became more positive after watching the commercial t (11) = 4.39, p < .05 t (degrees of freedom) = t obs
, p < place the level of significance here.
SPSS will give you the exact write p - value and in that case, you would
= .0025 or whatever # was computed.
Sometimes the p value is so small that SPSS computes the p value to be .0000000
A p p
value can never be 0. I repeat, a something like that. p value can never be 0.
As the research author, you would simply write p < .001 or
One Sample T Test Step by Step

T-Test Chp 9 and 10 (Spg 2005) 15
1: Restate the Question as a Research Hypothesis / Alternative Hypoth (H
1
) and a Null
Hypoth (H o
) about the Population
Pop 1:
Pop 2:
Hypotheses written out:
H o
:
H
1
:
Hypotheses using statistical notation
For a nondirectional one sample t test:
H o
: μ
1
= μ
2
H
1
: μ
1
≠ μ
2
2: Determine the Characteristics of the Comparison Distribution
You will be given the pop mean, but you may or may not be given the pop SD (variance information).
If you do know the pop SD (or variance), then you use a Z distribution.
If you do NOT know the pop SD (or variance), then you use a t distribution.
You then have to estimate the pop SD from what you know of your sample's SD.
μ = (Here mu refers to the pop mean) s = (Here s is the sample standard deviation) s =
X
(Here s refers to the standard error of the mean; it also
X refers to the standard deviation of the distribution of means; it is abbreviated SE in journal articles) s s =
X n
3: Determine the Cutoff Sample Score (Critical Value) on the Comparison Distribution
(Sampling Dist) at Which the Null Hypoth Should Be Rejected.
What three pieces of information (order doesn't matter) do you need to know to complete this step?
1. Degrees of Freedom (n – 1):
2. Directionality of hypothesis (one-tailed or two-tailed; recall that two-tailed is more conservative):
3. Significance Level (alpha; α) ; (typically we set our alpha level at .05 or .01):
The information obtained from this step is what will allow you to shade the rejection region(s).
If the hypothesis test is two-tailed, your t cv
will be + . t cv
= the critical value(s) of t . You will plot this number(s) on the test statistic distribution (-2, -1, 0,
1, 2).
4. Determine Your Sample’s Score on the Comparison Dist

T-Test Chp 9 and 10 (Spg 2005) 16
Recall the comparison distribution is a type of distribution used in hypothesis testing. It represents the pop situation if the null hypothesis is true. It's the distribution to which you compare the score based on your sample's results.
When you sketch the comparison distribution, you need to place the pop mean in the middle of your distribution. You will demarcate* this distribution by using the standard error ( s ).
X
*Demarcate: to set or mark the limits of; delimit; to mark the difference btwn; distinguish; separate. t obs

X
 s
X

X is your sample mean.
μ is your pop mean. s is the standard error based on your sample's SD and sample size.
X t obs
refers to the observed value of the t test statistic. t obs
= plot this number on the comparison distribution sketch on the t test number line
(-2, -1, 0, 1, 2).
End
The t obs
should line up with the sample mean ( X ) on the null hypothesis distribution (where you placed the pop mean in the middle and demarcated the numberline by the SE).
5: Decide Whether to Reject the Null Hypoth or Retain the Null Hypoth
Does the observed value of t exceed the critical value(s) of t ? If so, then reject the null hypothesis and accept the alternative hypothesis.