Hypothesis Test for Two Independent Proportions

Hypothesis Test for two Independent Proportions  answers a research question through statistical analysis about two independent proportions.  This hypothesis test looks at whether or not there is a significant difference between the two proportion variables.  Using this hypothesis test our test statistic will be Z, and we will be therefore be modeling with a Null Distribution.  This hypothesis test has five steps that I will go through.

1. Create you hypothesis:

You will have state a null hypothesis which will be what you expect to occur. You should write your null hypothesis in sentence form and in a mathematical formula; as shown below.

H_o: p_1 - p_2 = 0  Hence the probability of both variables success rate is the same.

Your alternate hypothesis can be written in three different forms depending how you want to test your data. Depending if you’d like to test your data with a two-tailed test, or a one-tailed test.  You should also write your null hypothesis in sentence form, and also in a mathematical formula; as shown below.

Case (1):H_{a}: [p_{1} - p_{2} \neq 0]

Case(2): H_{a}: [p_{1} - p_{2}> 0]

Case(3): H_{a}: [p_{1} - p_{2}< 0]

2.Check Sampling Conditions

The first condition you should check is to make sure that your sample was randomly selected.  You also want to check the independence condition to ensure that your sample is not more than 10% of the total population. If those conditions are met you then want to check the Success/Failure condition. When you are using this specific hypothesis test we will pool our success and failure values for each group.  Pooling in shown in the equations below.

\hat{p}_{pooled} = \frac{n_{1} \hat{p}_{1} +n_{2} \hat{p}_{2}}{n_{1} + n_{2}} (for successes)

\hat{q}_{pooled} = \frac{n_{1} \hat{q}_{1} +n_{2} \hat{q}_{2}}{n_{1} + n_{2}} (for failures)

Success Failure Condition (must pass each case):

n_1\cdot \hat{p}_{pooled} \geq 10    n_2\cdot \hat{p}_{pooled} \geq 10

n_1\cdot \hat{q}_{pooled} \geq 10      n_2\cdot \hat{q}_{pooled} \geq 10

3. Calculate your test statistic.  Your test statistic is a numerical summary of a data set, that reduces the data to one value; which can be used to preform a hypothesis test it says here. Equation for test statistic:

$latex Z = \frac{ (\hat{p}_{1}  – \hat{p}_{2}) – (p_{1} – p_{2})}{ \sqrt{ \frac{\hat{p}_{pooled} \cdot \hat{q}_{pooled}}{n_{1}}+\frac{\hat{p}_{pooled} \cdot \hat{q}_{pooled}}{n_{2}}}}$

4. Sketch a normal model & find your p-value.

Using our test statistic Z we can use SPSS and find our p-value.  With these pieces of information we can sketch a Normal distribution.

5. Conclusion

If your p-value < than \alpha you can reject your null hypothesis.  If your p-value > than \alpha you will report that you have failed to reject your null hypothesis.  You must make sure you express this in a contextual meaningful way with regards to what your research is dealing with.

You can find more information and help with using this test here. As pooling is specific to this type of hypothesis test, here is a link that provides more information on pooling and when to use it.




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