Wiki Page, Math 13 - Perez Sandbox : Module 21 - Wrap Up "Hypothesis Test for a Population Proportion"-2

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2026-01-17T21:15:34-08:00

Let’s Summarize

In this section, we looked at the four steps of a hypothesis test as they relate to a claim about a population proportion.

Step 1: Determine the hypotheses.

The hypotheses are claims about the population proportion, p.

The null hypothesis is a hypothesis that the proportion equals a specific value, .

The alternative hypothesis is the competing claim that the parameter is less than, greater than, or not equal to .

Step 2: Collect the data.

Since the hypothesis test is based on probability, random selection (and random assignment in an experiment) is essential in data production.

If you are not using a simulation, then verify that the normal curve can represent the distribution of sample proportions: np₀ and n(1 – p₀) must both be at least 10.

Step 3: Assess the evidence.

Conduct the simulation or determine the test statistic which is the z-score for the sample proportion. The formula is:

If using the normal model, find the P-value using StatCrunch or the Normal Distribution Calculator.

If the alternative hypothesis is greater than, the P-value is the area to the right of the test statistic. If the alternative hypothesis is less than, the P-value is the area to the left of the test statistic. If the alternative hypothesis is not equal to, the P-value is equal to double the tail area beyond the test statistic.

Step 4: Give the conclusion.

A small P-value says the data is unlikely to occur if the null is true. If the P-value is less than or equal to the significance level, we reject the null hypothesis and accept the alternative hypothesis instead.

If the P-value is greater than the significance level, we say we “fail to reject” the null hypothesis. We never say that we “accept” the null hypothesis. We just say that we don’t have enough evidence to reject it. This is equivalent to saying we don’t have enough evidence to support the alternative hypothesis.

We write the conclusion in the context of the research question. Our conclusion is usually a statement about the alternative hypothesis (we accept or fail to accept ) and should include the P-value.

Other Hypothesis Testing Notes

Remember that the P-value is the probability of seeing a sample proportion as extreme as the one observed from the data if the null hypothesis is true. The probability is about the random sample, not about the null or alternative hypothesis.

A larger sample size makes it more likely that we will reject the null hypothesis if the alternative is true. Another way of thinking about this is that increasing the sample size will decrease the likelihood of a Type II error. Recall that a Type II error is failing to reject the null hypothesis when the alternative is true.

Increasing the sample size can have the unintended effect of making the test sensitive to differences so small they don’t matter. A statistically significant difference is one large enough that it is unlikely to be due to sampling variability alone. Even a difference so small that it is not important can be statistically significant if the sample size is big enough.

Finally, remember the phrase “garbage in, garbage out.” If the data collection methods are poor, then the results of a hypothesis test are meaningless. No statistical methods can create useful information if our data comes from convenience or voluntary response samples. Additionally, the results of a hypothesis test apply only to the population from whom the sample was chosen.

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