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Hypothesis Testing for Proportions Study Guide (page 2)

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Updated on Oct 5, 2011

Example

A farmer wants to determine whether or not his field needs to be treated to control the number of insects in it. Because of costs and environmental concerns, he wants to be sure that treatment is required before proceeding. Based on this, answer the following questions.

  1. State the null and alternative hypotheses.
  2. How could the farmer make a type I error and what would the consequences be?
  3. How could the farmer make a type II error and what would the consequences be?

Solution

  1. Because the farmer wants to be certain that treatment is necessary before treating to control insects, this action (of treatment) is the alternative hypothesis. Thus, the null hypothesis is H0: Do not treat the field, and the alternative hypothesis is Ha: Treat the field.
  2. A type I error would occur if the farmer treated the field when it should not have been treated (rejected a true null hypothesis). This would cause him to spend money unnecessarily on treatment, reducing his profits for the season. The potential for negative environmental impacts is also present.
  3. A type II error would occur if the farmer did not treat the field when it should have been treated (failed to reject a false null hypothesis). This would result in lower production and thus a reduction in profits.

Conducting Hypothesis Tests on Proportions

We will follow five steps in conducting a hypothesis test. Each of these steps will be discussed and applied to proportions in this section.

Step 1: Specifying the Hypotheses

In science, a research hypothesis is a specific, testable prediction made about outcomes of a study. The scientist hopes that the results of the study validate the prediction. When establishing a set (H0 and Ha) of statistical hypotheses, the research hypothesis is made the alternative hypothesis. To understand why, think back to the parallel we have been using with a jury trial. If the jury believes that strong evidence exists against the assumption of not guilty, then they reject that assumption and conclude that the defendant is guilty. It is not necessary to prove innocence, it is only necessary to raise doubt about guilt to conclude not guilty. Similarly, with statistical hypotheses, if we reject the null hypothesis, we accept the alternative hypothesis. If sufficient evidence does not exist to reject the null hypothesis, we do not accept the null; we fail to reject it, which is a much weaker conclusion.

When working with proportions, the null hypothesis is that the population proportion p is equal to some proportion p0. The alternative may be that p is less than, greater than, or equal to p0, depending on what the research hypothesis is.

Step 2: Verify Necessary Conditions for a Test and, if Satisfied, Construct the Test Statistic

The conditions for testing hypotheses about the population proportion p are the same as those for constructing a confidence interval on this parameter. They are (1) the sample was randomly selected and (2) the sample is sufficiently large, that is, np ≥ 10 and n(1 – p) ≥ 10.

By the Central Limit Theorem, if n is sufficiently large, the sample proportion is approximately normally distributed with mean p and standard deviation . Standardizing , we have is approximately standard normal. If the null hypothesis is true and p = p0, the test statistic is approximately distributed as a standard normal random variable. Note: The test statistic is always constructed assuming that the null hypothesis is true.

Notice that the test statistic has the form .

The test statistics we will encounter in this book all have this form. They are standardized random variables whose distributions we know if the null hypothesis is true.

Step 3: Find the p-Value Associated with the Test Statistic

If the null hypothesis is true, the test statistic has an approximate standard normal distribution. If the null hypothesis is not true, the test statistic is not distributed as an approximate standard normal and is more likely to assume a value that is "unusual" for a random observation from a standard normal. The p–value is the probability of determining the probability of observing a value as extreme or more extreme as zT from a random selection of the standard normal distribution.

How do we measure how unusual a test statistic is? It depends on the alternative hypothesis. These are summarized in Figures 16.1, 16.2, and 16.3.

Figure 16.1

Figure 16.2

Figure 16.3

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