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Percents menu
Percents are understood by nearly
everyone, and therefore, they are the
most popular statistics cited in
research. Researchers are often
interested in comparing two percentages
to determine whether there is a
significant difference between them.
The Percents menu has three
selections:
Choosing the proper test
There are two kinds of t-tests between
percents. Which test you use depends upon
whether you're comparing percentages from
one or two samples.
Every percentage can be expressed as a
fraction. By looking at the denominator
of the fraction we can determine whether
to use a one-sample or two-sample t-test
between percents. If the denominators
used to calculate the two percentages
represent the same people, we use a
one-sample t-test between percents to
compare the two percents. If the
denominators represent different people,
we use the two-sample t-test between
percents.
For example suppose you did a survey
of 200 people. Your survey asked,
Were you satisfied with the
program?
___ Yes ___ No ___ Don't know
Of the 200 people, 80 said yes, 100
said no, and 20 didn't know. You could
summarize the responses as:
Yes 80/200 = .4 = 40%
No 100/200 = .5 = 50%
Don't know 20/200 = .1 = 10%
Is there a significant difference
between the percent saying yes (40%) and
the percent saying no (50%)? Obviously,
there is a difference; but how sure are
we that the difference didn't just happen
by chance? In other words, how reliable
is the difference?
Notice that the denominator used to
calculate the percent of yes responses
(200) represents the same people as the
denominator used to calculate the percent
of no responses (200). Therefore, we use
a one-sample t-test between proportions.
The key is that the denominators
represent the same people (not that they
are the same number).
After you completed your survey,
another group of researchers tried to
replicate your study. They also used a
sample size of 200, and asked the
identical question. Of the 200 people in
their survey, 60 said yes, 100 said no,
and 40 didn't know. They summarized their
results as:
Yes 60/200 = .3 = 30%
No 100/200 = .5 = 50%
Don't know 40/200 = .2 = 20%
Is there a significant difference
between the percent who said yes in your
survey (40%) and the percent that said
yes in their survey (30%)? For your
survey the percent that said yes was
calculated as 80/200, and in their survey
it was 60/200. To compare the yes
responses between the two surveys, we
would use a two-sample t-test between
percents. Even though both denominators
were 200, they do not represent the same
200 people.
Examples that would use a one-sample t-test
Which proposal
would you vote for?
___ Proposal A
___ Proposal B
Which product do
you like better?
___ Name Brand
___ Brand X
Which candidate
would you vote for?
___ Johnson ___
Smith ___ Anderson
When there are
more than two choices, you can do the
t-test between any two of them. In this
example, there are three possible
combinations: Johnson/Smith,
Johnson/Anderson, and Smith/Anderson.
Thus, you could actually perform three
separate t-tests...one for each pair of
candidates. If this was your analysis
plan, you would also use Bonferroni's
theorem to adjust the critical alpha
level because the plan involved multiple
tests of the same type and family.
Examples
that would use a two-sample t-test
A previous study
found that 39% of the public believed in
gun control. Your study found the 34%
believed in gun control. Are the beliefs
of your sample different than those of
the previous study?
The results of a
magazine readership study showed that 17%
of the women and 11% of the men recalled
seeing your ad in the last issue. Is
there a significant difference between
men and women?
In a brand
awareness study, 25% of the respondents
from the Western region had heard of your
product. However, only 18% of the
respondents from the Eastern region had
heard of your product. Is there a
significant difference in product
awareness between the Eastern and Western
regions?
One sample t-test between percents
This test can be
performed to determine whether
respondents are more likely to prefer one
alternative or another.
Example
The research
question is: Is there a significant
difference between the percent of people
who say they would vote for candidate A
and the percent of people who say they
will vote for candidate B? The null
hypothesis is: There is no significant
difference between the percent of people
who say they will vote for candidate A or
candidate B. The results of the survey
were:
Plan to vote for
candidate A = 35.5%
Plan to vote for
candidate B = 22.4%
Sample size =
107
The sum of the
two percents does not have to be equal to
100 (there may be candidates C and D, and
people that have no opinion). Use a
one-sample t-test because both
percentages came from a single sample.
Use a two-tailed
probability because the null hypothesis
does not state the direction of the
difference. If the hypothesis is that one
particular choice has a greater
percentage, use a one-tailed test (divide
the two-tailed probability by two).
Enter the first
percent: 35.5
Enter the second percent: 22.4
Enter the sample size: 107
-----------------------------------------
t-value = 1.808
Degrees of freedom = 106
Two-tailed probability = .074
You might make a
statement in a report like this: A
one-sample t-test between proportions was
performed to determine whether there was
a significant difference between the
percent choosing candidate A and
candidate B. The t-statistic was not
significant at the .05 critical alpha
level, t(106)=1.808, p=.073. Therefore,
we fail to reject the null hypothesis and
conclude that the difference was not
significant.
Two sample t-test between percents
This test can be used to compare
percentages drawn from two independent
samples. It can also be used to compare
two subgroups from a single sample.
Example
After conducting
a survey of customers, you want to
compare the attributes of men and women.
Even though all respondents were part of
the same survey, the men and women are
treated as two samples. The percent of
men with a particular attribute is
calculated using the total number of men
as the denominator for the fraction. And
the percent of women with the attribute
is calculate using the total number of
women as the denominator. Since the
denominators for the two fractions
represent different people, a two-sample
t-test between percents is appropriate.
The research
question is: Is there a significant
difference between the proportion of men
having the attribute and the proportion
of women having the attribute? The null
hypothesis is: There is no significant
difference between the proportion of men
having the attribute and the proportion
of women having the attribute. The
results of the survey were:
86 men were
surveyed and 22 of them (25.6%) had the
attribute.
49 women were
surveyed and 19 of them (38.8%) had the
attribute.
Enter the first
percent: 25.6
Enter the sample size for the first
percent: 86
Enter the second percent: 38.8
Enter the sample size for the second
percent: 49
-------------------------------------------------------------
t-value = 1.603
Degrees of freedom = 133
Two-tailed probability = .111
You might make a
statement in a report like this: A
two-sample t-test between proportions was
performed to determine whether there was
a significant difference between men and
women with respect to the percent who had
the attribute. The t-statistic was not
significant at the .05 critical alpha
level, t(133)=1.603, p=.111. Therefore,
we fail to reject the null hypothesis and
conclude that the difference between men
and women was not significant.
Another example
Suppose
interviews were conducted at two
different shopping centers. This two
sample t-test between percents could be
used to determine if the responses from
the two shopping centers were different.
The research
question is: Is there a significant
difference between shopping centers A and
B with respect to the percent that say
they would buy product X? The null
hypothesis is: There is no significant
difference between shopping centers A and
B with respect to the percent of people
that say they would buy product X. A
two-tailed probability will be used
because the hypothesis does not state the
direction of the difference. The results
of the survey were:
89 people were
interviewed as shopping center A and 57
of them (64.0%) said they would buy
product X.
92 people were
interviewed as shopping center B and 51
of them (55.4%) said they would buy
product X.
Enter the first
percent: 64.0
Enter the sample size for the first
percent: 89
Enter the second percent: 55.4
Enter the sample size for the second
percent: 92
-------------------------------------------------------------
t-value = 1.179
Degrees of freedom = 179
Two-tailed probability = .240
You might write
a paragraph in a report like this: A
two-sample t-test between proportions was
performed to determine whether there was
a significant difference between the two
shopping centers with respect to the
percent who said they would buy product
X. The t-statistic was not significant at
the .05 critical alpha level,
t(179)=1.179, p=.240. Therefore, we fail
to reject the null hypothesis and
conclude that the difference in responses
between the two shopping centers was not
significant.
Confidence intervals around a percent
Confidence intervals are used to
determine how much latitude there is in
the range of a percent if we were to take
repeated samples from the population.
Example
In a study of
150 customers, you find that 60 percent
have a college degree. Your best estimate
of the percent who have a college degree
in the population of customers is also 60
percent. However, since it is just an
estimate, we establish confidence
intervals around the estimate as a way of
showing how reliable the estimate is.
Confidence
intervals can be established for any
error rate you are willing to accept. If,
for example, you choose the 95%
confidence interval, you would expect
that in five percent of the samples drawn
from the population, the percent who had
a college degree would fall outside of
the interval.
What are the 95%
confidence intervals around this percent?
In the following example, note that no
value is entered for the population size.
When the population is very large
compared to the sample size (as in most
research), it is not necessary to enter a
population size. If, however, the sample
represents more than ten percent of the
population, the formulas incorporate a
finite population correction adjustment.
Thus, you only need to enter the
population size when the sample size
exceeds ten percent of the population
size.
Enter the
percent: 60
Enter the sample size: 150
Enter the population size: (left blank)
Enter the desired confidence interval
(%): 95
----------------------------------------------------------
Standard error
of the proportion = .040
Degrees of freedom = 149
95% confidence interval = 60.0%  7.9%
Confidence interval range = 52.1% to
67.9%
Therefore, our
best estimate of the population
proportion with 5% error is 60%  7.9%. Stated differently, if
we predict that the proportion in the
population who have a college degree is
between 52.1% and 67.9%, our prediction
would be wrong for 5% of the samples that
we draw from the population.
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