# Testing Hypotheses about the Difference Between the Means of

Document Sample

```					Testing Hypotheses about the
Difference Between the Means of
Two Populations

STAT Chapter 9
METH Chapter 13

S07

Review: Hypothesis testing steps
1.     Choose the appropriate test (statistical model)
2.     Create the Null and Alternative Hypotheses
3.     State the criteria for rejecting Null
4.     Do the calculations
5.     Make the decision: Reject Null or Not
6.     Draw Conclusions
7.     Write the results up in an APA-style format

(Steps 1, 2, & 3 constitute “Setting up the Hypothesis
Test”)

Example:
Recently, an endogenous brain neurotransmitter (NT)
called Galanin has been discovered that appears to
specifically affect one’s desire to eat foods with high fat
content. The more of this naturally occurring NT that an
individual possesses, the higher is his or her craving for
high-fat foods. A drug company developed a drug that
blocks Galanin without affecting the appetite for healthier
(less fat) foods. They hope it will be useful for obesity.
After approval by the FDA, one of the neuroscientists of
the drug company conducted an experiment in which 15
obese female volunteers were randomly selected and given
the experimental drug for 6 months. Baseline and ending (6
month) weights were recorded for each subject.

1
Subject Baseline        Ending
The data for            Number Weight           Weight
1      165            145
the example:               2      143            137
3      175            170
4      135            136
5      148            141
6      155            138
7      158            137
8      140            125
9      172            161
10      164            156
11      178            165
12      182            170
13      190            176
14      169            154
15      157            143

REVIEW: Hypothesis testing Step 1
Choose the sampling distribution or statistical
model
First, ask what type of data: nominal or
interval/ratio?
Second, ask what question are you interested in

Our answers (at the moment) are interval/ratio
data and differences: therefore, z, t, or F tests

(see Flowchart for choosing correct statistics)

Review: Step 1 continued

Next question is: How many conditions (or
samples or groups)?
If you have 1 sample of data, then you must
choose between a z-test or a one-sample t-test
If there are 2 conditions (or 2 samples of
data), then you must choose between the two
types of two-sample t-tests
If there are 3 conditions/samples (or more),
then you must choose the appropriate F-test or
ANOVA (we’ll cover these tests later ☺)

2
One-sample hypothesis tests compare one
sample mean to a population mean, in order to
determine the likelihood that the sample is part of that
population, or whether it has been drawn from a
different population (are they statistically different?)

z-tests:                    t-tests:
Must know μ (mean)           Must know μ (mean) of
and σ (standard              the population data
deviation) of the            Use s (standard
population data              deviation of the sample
data) to estimate the
standard error of the
mean

Two-sample t-tests are used to
determine whether the means of two groups of
scores differ from each other to a statistically
significant degree

Whenever there is more than 1 condition, or
sample of data, the next question to ask is
what type of subject design was used?
Independent groups
vs.
Repeated measures

There are 2 types of two-sample t-tests

One is used for          One is used for these
these designs:           designs:
Random groups             Repeated measures
designs                   designs
Natural groups            Matched groups designs
designs
Multiple names:
Multiple names                Repeated measures
Independent               Related samples
groups                    Correlated samples
Independent               Dependent samples
samples                   Paired samples
Matched samples

3
Hypothesis testing: Step 1
What statistical test do we use?

Two-       t-
Two-sample t-test

Which one?

Repeated measures

Review: Hypothesis testing steps
1.   Choose the appropriate test (statistical model)
2.   Create the Null and Alternative Hypotheses
3.   State the criteria for rejecting Null
4.   Do the calculations
5.   Make the decision: Reject Null or Not
6.   Draw Conclusions
7.   Write the results up in an APA-style format

(Steps 1, 2, & 3 constitute “Setting up the Hypothesis
Test”)

Step 2: Null and alternative
hypotheses?
Null
H0: µ1 = µ2
H0: µD = 0

Alternative
H1: µ1 ≠ µ2
H1: µD ≠ 0

4
Step 3: Criteria for rejecting Null?
Use standard choices for Psychologists:
α = .05
Two-tailed test

Determine critical value of t:
df = N – 1, where N is the number of pairs
df = 15 – 1 = 14
Check Appendix in your textbook for the critical
value of t corresponding to the above info:
tcritical = 2.145

Step 4: Do the calculations

Enter the data into SPSS
CRITICAL POINT:
If more than 1 data point is obtained from each subject (or
when you’ve matched subjects), all the data goes on the
same row for a given subject and each column is labeled
If there are 2 samples of data, but each subject only gives
1 piece of data, then you have 1 column for the data itself,
and a second column to indicate which condition the
subject is in

Subject Baseline              Ending
The data for              Number Weight                 Weight
1      165                  145
the example:                 2      143                  137
3      175                  170
4      135                  136
5      148                  141
6      155                  138
7      158                  137
8      140                  125
9      172                  161
10      164                  156
11      178                  165
12      182                  170
13      190                  176
14      169                  154
15      157                  143

5
Step 4: Do the calculations, cont’d.

Double-check to make sure that the data was
entered correctly

Click on Analyze
Then Compare Means,
Then Paired-Samples t-test,
Click on each variable to be analyzed in
this test, then click on the arrow in the
middle of the two boxes
Click on OK

SPSS output
Paired Samples Statistics

Std. Error
Mean            N        Std. Deviation     Mean
Pair     baseline      162.0667             15        16.10442     4.15814
1        endweight     150.2667             15        15.42478     3.98266

Paired Samples Test

Paired Differences
95% Confidence
Interval of the
Std. Error    Difference
Mean Std. Deviation Mean Lower Upper              t     df        Sig. (2-tailed
Pair 1 baseline - endwe
1.80000    5.93055 1.53126 8.51577 5.08423        7.706        14           .000

Step 5: Make the decision
Compare the tobtained (just calculated) to the
tcritical in order to make the decision:

tobtained = 7.706 and tcritical = 2.145

Is tobtained larger than tcritical ?

Yes. Reject the null.

6
Step 6: Draw conclusions
There is a significant difference between baseline
weight and ending weight.

Yes.

The average baseline weight was 162.07 and the
average ending weight was 150.27. Therefore, the
women on the drug lost an average of 11.8 lbs.

You’ll need a graph:
Click on Graphs
Choose Bar
Choose Simple, and at the bottom, click on
Summaries of separate variables, then click on
Define
Click on both variables on the left, then click on
the arrow, then choose OK
Go into Chart Editor (by double-clicking
somewhere on the Graph itself), and re-
format the graph in APA style
X out of Chart editor, then copy and paste
the graph into a word document

Step 7: Write it up in APA-style
A paired-samples t-test was
used to compare the
participants’ weight before
beginning (baseline) and again
after 6 months on the drug that
was purported to block
cravings for high-fat foods.
The participants lost a
significant amount of weight
(11.8 lbs) on the drug across
the six months, t(14) = 7.71, p
< .001. The data are graphed in
Figure 1.

7
Example 2
A psychologists is interested in determining whether
immediate memory capacity is affected by sleep loss.
Immediate memory is defined as the amount of material that can
be remembered immediately after it has been presented. Twelve
students are randomly selected from Introductory Psychology
and randomly assigned to 2 groups. One of the groups is sleep-
deprived for 24 hours before the material is presented. All
subjects in the other group receive the normal amount of sleep
(7-8 hours). The material consists of a series of slides, with
each containing nine numbers. Each slide is presented for a
short time interval (50 ms) after which the subject must recall as
many numbers as possible. The scores represent the percentage
correctly recalled.

Review: Hypothesis testing steps
1.   Choose the appropriate test (statistical model)
2.   Create the Null and Alternative Hypotheses
3.   State the criteria for rejecting Null
4.   Do the calculations
5.   Make the decision: Reject Null or Not
6.   Draw Conclusions
7.   Write the results up in an APA-style format

(Steps 1, 2, & 3 constitute “Setting up the Hypothesis
Test”)

Hypothesis testing: Step 1
What statistical test do we use?

Two-       t-
Two-sample t-test

Which one?

Independent samples

8
Step 2: Null and alternative
hypotheses?

Null
H0: µ1 = µ2                or       X1 = X2

Alternative
H1: µ1 ≠ µ2               or       X1 ≠ X2

Step 3: Criteria for rejecting Null?
Use standard choices for Psychologists:
α = .05
Two-tailed test

Determine critical value of t:
df = N1 + N2 – 2
df = 12 – 2 = 10
Check Appendix in your textbook for the critical
value of t corresponding to the above info:
tcritical = 2.228

Step 4: Do the calculations

Enter the data into SPSS
CRITICAL POINT:
If more than 1 data point is obtained from each subject (or when
you’ve matched subjects), all the data goes on the same row for
a given subject and each column is labeled
If there are 2 samples of data, but each subject only gives 1
piece of data, then you have 1 column for the data itself, and a
second column to indicate which condition the subject is in

9
Data for                                           Subject                     Group               Percent
Example 2:                                         Number                                          Recalled
1                            1                 68
2                            1                 73
3                            1                 72
Group 1 =                                      4                            1                 65
non-deprived                                   5                            1                 70
control group                                  6                            1                 73
7                            2                 70
8                            2                 64
Group 2 =                                      9                            2                 68
sleep-deprived                                10                            2                 63
11                            2                 69
group
12                            2                 66

Step 4: Do the calculations, cont’d.
Double-check to make sure that the data was entered
correctly
Choose Analyze
Click on Compare Means
Choose Independent-Samples t-test,
Put column with data in Test Variable box
and column with group in Grouping
Variable box
Click on Define Groups, enter correct
numbers
Then click on OK

SPSS output
Group Statistics

Std. Error
1=control,2=no sleep           N             Mean        Std. Deviation        Mean
percent   1.00                                 6       70.1667           3.18852        1.30171
2.00                                 6       66.6667           2.80476        1.14504

Independent Samples Test

Levene's Test for
Equality of Variances                              t-test for Equality of Means
95% Confidence
Interval of the
Mean     Std. Error      Difference
F       Sig.         t          df        Sig. (2-tailed) Difference Difference   Lower      Upper
percent Equal variances
.042      .842      2.019            10           .071    3.50000     1.73365     -.36282   7.36282
assumed
Equal variances
2.019       9.840             .072    3.50000     1.73365     -.37136   7.37136
not assumed

10
Step 5: Make the decision
Compare the tobtained (just calculated) to the
tcritical in order to make the decision:

Is tobtained larger than tcritical ?

tobtained = 2.019 vs. tcritical = 2.228

No. Do not reject the null.

Step 6: Draw conclusions
There is not a significant difference in percent of
numbers remembered between subjects that were
sleep deprived versus those that were not.

Yes.

Non-sleep-deprived subjects remembered an
average of 70.17%, whereas sleep-deprived
subjects remembered an average of 66.67% of the
numbers.

You’ll want a graph of the data…
Click on Graphs
Choose Simple, then click on Summaries for groups of cases (at
bottom of window), then Define
Click on Other Statistic (e.g., mean) under Bars Represent
box
Then choose your data column on the left, and click on the
arrow in the Bars Represent box to move it in as the Variable
Click on the group column on the left, then click on the
arrow to move it into the Category Axis box
Click on OK
Go into Chart Editor and format it for APA style
Copy and paste it into your word document

11
Step 7: Write it up in APA-style
The percentage of numbers
recalled correctly for sleep-
deprived versus non-sleep-
deprived students was analyzed
with an independent-samples t-
test. The percent recalled for
the sleep-deprived participants
did not differ significantly
(t(10) = 2.02, p = .071) from
the percent recalled by the
participants that slept the
normal amount. As can be
seen in the graph (see Figure
1), the two groups did not
differ.

Underlying Assumptions for t-tests
In theory:
The data should be normally distributed within each
population
The variances of the two populations should be equal

In practice:
You can generally ignore the assumption of normally
distributed data in the population
You can ignore the assumption about equal variances as
well, as long as the number of subjects in each
condition is approximately equal

Confidence Limits on MD
Paired-samples t-test:
XD ± (tcritical) ( sD)

Where:

MD = mean of the difference scores
sD = standard error of the mean of the
differences

12
Confidence Limits on µ1 - µ2
Independent-samples t-test

(X1 – X2) ± (tcritical) ( s x1-x2)

Standard error of the difference between means

= (N1 – 1) s12 + (N2 – 1) s22 (1/N1 + 1/N2)
√      N1 + N2 - 2

Confidence limits for the 2 examples

Example 1:                        Example 2:

11.8 ± (2.145) (1.53126)          (70.1667 – 66.6667)
=                                 ± (2.228) (1.733652)

11.8 ± 3.2846 =                   = 3.5 ± 3.86258

8.5154 – 15.0846                  = -.3626 – 7.3626

13

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