The One and Two Sample Z proportion is used to test the proportion of one or two samples. It is a parametric test.



One Sample proportion

In one-sample Z proportion, we compare one sample with a target value.

For run this test it’s important that:

  • The sample come from a binomial distribution;
  • When we have mean (n*p) and variance (n*(1-p)) > 10, the binomial distribution can be approzimated with a standard normal distribution;


In this case the hypothesis are like (in case of two tail test):

  • H0: The population proportion is equal to a specific value;
  • H1The population proportion is different from a specific value;


The way to conduct the hypothesis test is like the one-sample t-tests, but we use another statistics.


In this test we use the Z statistic in the image1:

image1 - z statistic
image1 – z statistic

where:

  • p is the observed proportion in the sample;
  • p0 is the target proportion;
  • n is the number of observations of the sample;


Example: We have a coin toss experiment where we count the number of heads. Our sample is about ten flips where we look at eight heads. In this case we have
- P = 8/10 = 0,8
- P0 = 5/10 = 0,5 because we suppose a fair coin;
- n = 10

After we have computer our Z statistics, we need to look at the critical value from the Z table. Then we only need to compare the statistics with the critical value for reject or fail to reject H0.


Two Sample proportion

In two-sample Z proportion we compare the proportion between two sample.

The process is similar of the one sample proportion (using the standard distribution) but we use the statitics in image2:

image2 - Z statistics for two proportion
image2 – Z statistics for two proportion

where:

  • p1 and p2 are the proportion of the two sample;
  • n1 and n2 is the number of observation in the two sample;
  • p is the combined proportion of the two sample;


After we have computer our Z statistics, we need to look at the critical value from the Z table. Then we only need to compare the statistics with the critical value for reject or fail to reject H0.

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