9.1.2.1 - Normal Approximation Method Formulas

1. Check any necessary assumptions and write null and alternative hypotheses. Section

To use the normal approximation method a minimum of 10 successes and 10 failures in each group are necessary (i.e., \(n p \geq 10\) and \(n (1-p) \geq 10\)).

The two groups that are being compared must be unpaired and unrelated (i.e., independent).

Below are the possible null and alternative hypothesis pairs:

Research Question Are the proportions of group 1 and group 2 different? Is the proportion of group 1 greater than the proportion of group 2? Is the proportion of group 1 less than the proportion of group 2?
Null Hypothesis, \(H_{0}\) \(p_1 - p_2=0\) \(p_1 - p_2=0\) \(p_1 - p_2=0\)
Alternative Hypothesis, \(H_{a}\) \(p_1 - p_2 \neq 0\) \(p_1 - p_2> 0\) \(p_1 - p_2<0\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
2. Calculate an appropriate test statistic.

The null hypothesis is that there is not a difference between the two proportions (i.e., \(p_1 = p_2\)). If the null hypothesis is true then the population proportions are equal. When computing the standard error for the difference between the two proportions a pooled proportion is used as opposed to the two proportions separately (i.e., unpooled). This pooled estimate will be symbolized by \(\widehat{p}\). This is similar to a weighted mean, but with two proportions. 

Pooled Estimate of \(p\)
\(\widehat{p}=\dfrac{\widehat{p}_1n_1+\widehat{p}_2n_2}{n_1+n_2}\)

The standard error for the difference between two proportions is symbolized by \(SE_{0}\). The subscript 0 tells us that this standard error is computed under the null hypothesis (\(H_0: p_1-p_2=0\)).

Standard Error

\(SE_0={\sqrt{\dfrac{\widehat{p} (1-\widehat{p})}{n_1}+\dfrac{\widehat{p}(1-\widehat{p})}{n_2}}}=\sqrt{\widehat{p}(1-\widehat{p})\left ( \dfrac{1}{n_1}+\dfrac{1}{n_2} \right )}\)

Note that the default in many statistical programs, including Minitab Express, is to estimate the two proportions separately (i.e., unpooled). In order to obtain results using the pooled estimate of the proportion you will need to change the method.

Also note that this standard error is different from the one that you used when constructing a confidence interval for \(p_1-p_2\). While the hypothesis testing procedure is based on the null hypothesis that \(p_1-p_2=0\), the confidence interval approach is not based on this premise. The hypothesis testing approach uses the pooled estimate of \(p\) while the confidence interval approach will use an unpooled method. 

Test Statistic for Two Independent Proportions
\(z=\dfrac{\widehat{p}_1-\widehat{p}_2}{SE_0}\)
3. Determine the p value associated with the test statistic.

The \(z\) test statistic found in Step 2 is used to determine the \(p\) value. The \(p\) value is the proportion of the \(z\) distribution (normal distribution with a mean of 0 and standard deviation of 1) that is more extreme than the test statistic in the direction of the alternative hypothesis. 

4. Decide between the null and alternative hypotheses.

If \(p \leq \alpha\) reject the null hypothesis. If \(p>\alpha\) fail to reject the null hypothesis.

5. State a "real world" conclusion.

Based on your decision in Step 4, write a conclusion in terms of the original research question.