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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

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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

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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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