1 proportion test minitab example

Hypotheses Testing about Proportions

Hypotheses for 1 proportion test

For 1 Proportion, the hypotheses are:

Null hypothesis

H0: ρ = ρ0 The population proportion (ρ) equals the hypothesized

proportion (ρ0).

Alternative hypothesis

Choose one:

H1: ρ ≠ ρ0 The population proportion (ρ) differs from the

hypothesized proportion (ρ0).

H1: ρ > ρ0 The population proportion (ρ) is greater than the

hypothesized proportion (ρ0).

H1: ρ < ρ0 The population proportion (ρ) is less than the

hypothesized proportion (ρ0).

Methods that Minitab uses to calculate 1 proportion test

Minitab's 1 Proportion test uses the binomial distribution (also known as the exact method)

by default for calculating the hypothesis test and confidence interval. You can choose to

use a normal approximation by clicking Options, and choosing Normal approximation in

the dropdown beside Method. Most textbooks use the normal approximation method

because it is easy to calculate manually. However, the exact method is more accurate and

powerful than the normal approximation method.

With the normal approximation, it is possible to get different conclusions between the p-

value and the confidence interval. This is because this method uses the hypothesized

probability (ρ0) in the calculation of the test statistic and the observed probability in the

calculation of the confidence interval. This could cause a difference between the

conclusions, especially with larger values of Z which would indicate a large difference

between ρ0 and the population proportion.

Example 1: Suppose that there are 272 successes out of 800 trials. Test whether this sample

proportion is different from 0.37 at 10% significance level. Perform both exact test and

using normal approximation. Is there difference in conclusions? Perform the same test by

hand using normal approximation. Perform the same test using StatKey. Report p-value for

the last two tests.

Quiz question 1: Test whether the same number of successes out of the same number of

trials as in example 1 is greater than 0.37 at 10% significance level. Perform both exact test

and using normal approximation. Is there difference in conclusions? Perform the same test

by hand using normal approximation. Perform the same test using StatKey. Report p-value

for the last two tests.

You do not need to enter any data into the data file. After choosing ‘One Sample

Proportion’ in the field on the top choose ‘Summarized data’ and enter appropriate

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numbers into other fields. Be careful with the choice of the alternative hypothesis and

significance level.

Hypotheses for 2 proportions test

For 2 Proportions, the hypotheses are:

Null hypothesis

H0: ρ1- ρ2 = d0 The difference between the population proportions (ρ1-

ρ2) equals the hypothesized difference (d0).

Alternative hypothesis

Choose one:

H1: ρ1- ρ2 ≠ d0 The difference between the population proportions (ρ1-

ρ2) does not equal the hypothesized difference (d0).

H1: ρ1- ρ2 > d0 The difference between the population proportions (ρ1-

ρ2) is greater than the hypothesized difference (d0).

H1: ρ1- ρ2 < d0 The difference between the population proportions (ρ1-

ρ2) is less than the hypothesized difference (d0).

Methods that Minitab uses to calculate 2 proportions test

Minitab's 2 Proportions test uses a normal approximation by default for calculating the

hypothesis test and confidence interval. The normal approximation can be used to

approximate the difference between two binomial random variables provided the sample

sizes are large and proportions are not too close to 0% or 100%. In addition, when you

specify a test difference of zero in the Options sub-dialog box, Minitab does Fisher's exact

test, which is exact for all sample sizes and proportions. Fisher's exact test is based on the

hypergeometric distribution.

The normal approximation may be inaccurate for small numbers of events or nonevents. If

the number of events or nonevents in either sample is less than five, Minitab displays a

note. Fisher's exact test is accurate for all sample sizes and proportions.

Example 2: In clinical trials of a new allergy medicine, 3774 adult allergy patients were

randomly divided into 2 groups. The patients in group 1 (experimental group) received a

dose of the new medicine while patients in group 2 (control group) received a placebo. Of

the 2013 patients in the experimental group, 547 reported headaches as a side effect. Of the

1671 patients in the control group, 368 reported headaches as a side effect. Is there

significant evidence to conclude that the proportion of the new medicine users that

experienced headaches as a side effect is greater than the proportion in the control group at

the 5% significance level? Perform the same test by hand. Perform the same test using

StatKey. Report p-value for the last two tests.

Quiz question 2: Labor Day was created by the U.S. labor movement over 100 years ago. It

was subsequently adopted by most states as an official holiday. In a Gallup Poll, 1003

randomly selected adults were asked whether they approve of labor unions; 65% said yes.

a. In 1936, about 72% of Americans approved of labor unions. At the 5% significance

level, do the data provide sufficient evidence (at 5% significance level) to conclude that the

percentage of Americans who approve of labor unions now has decreased since 1936?

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Perform the same test by hand. Perform the same test using StatKey. Report p-value for the

last two tests.

b. In 1963, roughly 67% of Americans approved of labor unions. At the 5% significance

level, do the data provide sufficient evidence (at 5% significance level) to conclude that the

percentage of Americans who approve of labor unions now has decreased since 1963?

Perform the same test by hand. Perform the same test using StatKey. Report p-value for the

last two tests.