8.1.2.1 - Normal Approximation Method Formulas

Here we will be using the five step hypothesis testing procedure to compare the proportion in one random sample to a specified population proportion using the normal approximation method.

1. Check assumptions and write hypotheses

In order to use the normal approximation method, the assumption is that both \(n p_0 \geq 10\) and \(n (1-p_0) \geq 10\). Recall that \(p_0\) is the population proportion in the null hypothesis.

Research Question Is the proportion different from \(p_0\)? Is the proportion greater than \(p_0\)? Is the proportion less than \(p_0\)?
Null Hypothesis, \(H_{0}\) \(p=p_0\) \(p= p_0\) \(p= p_0\)
Alternative Hypothesis, \(H_{a}\) \(p\neq p_0\) \(p> p_0\) \(p< p_0\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional

Where \(p_0\) is the hypothesized population proportion that you are comparing your sample to.

2. Calculate the test statistic

When using the normal approximation method we will be using a z test statistic. The z test statistic tells us how far our sample proportion is from the hypothesized population proportion in standard error units. Note that this formula follows the basic structure of a test statistic that you learned last week: \(test\;statistic=\frac{sample\;statistic-null\;parameter}{standard\;error}\)

Test statistic: One Group Proportion
\(z=\frac{\widehat{p}- p_0 }{\sqrt{\frac{p_0 (1- p_0)}{n}}}\)

\(\widehat{p}\) = sample proportion
\(p_{0}\) = hypothesize population proportion
\(n\) = sample size

3. Determine the p-value

Given that the null hypothesis is true, the p value is the probability that a randomly selected sample of n would have a sample proportion as different, or more different, than the one in our sample, in the direction of the alternative hypothesis. We can find the p value by mapping the test statistic from step 2 onto the z distribution. 

Note that p-values are also symbolized by \(p\). Do not confuse this with the population proportion which shares the same symbol.

We can look up the \(p\)-value using Minitab Express by constructing the sampling distribution.  Because we are using the normal approximation here, we have a \(z\) test statistic that we can map onto the \(z\) distribution. Recall, the z distribution is a normal distribution with a mean of 0 and standard deviation of 1. If we are conducting a one-tailed (i.e., right- or left-tailed) test, we look up the area of the sampling distribution that is beyond our test statistic. If we are conducting a two-tailed (i.e., non-directional) test there is one additional step: we need to multiple the area by two to take into account the possibility of being in the right or left tail. 

4. Make a decision

We can decide between the null and alternative hypotheses by examining our p-value. If \(p \leq \alpha\) reject the null hypothesis. If \(p>\alpha\) fail to reject the null hypothesis. Unless stated otherwise, assume that \(\alpha=.05\).

When we reject the null hypothesis our results are said to be statistically significant.

5. State a "real world" conclusion

Based on our decision in step 4, we will write a sentence or two concerning our decision in relation to the original research question.