8.1.2.1 - Normal Approximation Method Formulas

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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 TestTwo-tailed, non-directionalRight-tailed, directionalLeft-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

Now, we use the test statistic that we computed in step 2 to determine the probability of obtaining a sample that deviates from the hypothesized population as much as or more than the sample that we have. In other words, given that the null hypothesis is true, the probability that a randomly selected sample of \(n\) would have a sample statistic as different as the one obtained (or more different) is the p-value.

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.