Wiki Page, Math 13 - Perez Sandbox : Module 24 - Hypothesis Test for a Population Mean (1a of 5)

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Hypothesis Test for a Population Mean: Introduction

In this module we learn to use a sample mean to test a hypothesis about a population mean.

We did hypothesis tests in earlier modules. We will see that the steps and the logic of the hypothesis test for a population mean are the same. As a reminder, here are the general steps in the process of hypothesis testing:

Step 1: Determine the hypotheses.

The hypotheses come from the research question. In this module the hypotheses involve a population mean, .

The null hypothesis says the population mean equals a number: : , where is a number.

The alternative hypothesis says the population mean is less than, greater than, or not equal to the number: : OR OR

Step 2: Collect the data.

We select a random sample from the population. The data comes from this sample. We calculate a statistic, in this case a sample mean, to summarize the data.

Step 3: Assess the evidence.

If we assume that the null hypothesis is true, could the data come from the population described by the null hypothesis? In other words, are sample means more extreme that the observed data likely (large P-value) or unlikely (small P-value)?

Use a simulation or a mathematical model to examine the results from random samples selected from the population described by the null hypothesis.

If we use a mathematical model, we will use the T-model if conditions are met. The test statistic is similar to a Z-score:

where the estimated standard error is .

Use the t-statistic with appropriate degrees of freedom to find the P-value.

As before, a hypothesis test can be one-tailed or two-tailed. If the alternative hypothesis involves a less than or greater than statement, then the P-value is the area in one tail: left tail for less than and right tail for greater than. If the inequality is not equals , the test is two-tailed and the P-value is the area in both tails.

Step 4: State a conclusion.

We use what we find in the previous step to make a decision. This step requires us to think in the following way. Remember that we assume that the null hypothesis is true. Then one of two outcomes can occur:

One possibility is that results similar to the actual sample are extremely unlikely (small P-value) when the null hypothesis is true.

This means that the data from the observed sample do not fit in with results from random samples selected from the population described by the null hypothesis. In other words, the difference between the observed sample mean and the hypothesized population mean (the error) is too large to attribute to chance (the expected variability that occurs in random sampling.)

In this case, it is unlikely that the data came from this population, so we view this as strong evidence against the null hypothesis. We reject the null hypothesis in favor of the alternative hypothesis.

The other possibility is that results similar to the actual sample are fairly likely (not unusual – large P-value).

This means that the data fit in with typical results from random samples selected from the population described by the null hypothesis. In other words, the difference between the observed sample mean and the hypothesized population mean (the error) is attributed to chance (the expected variability that occurs in random sampling.)

In this case, we do not have strong evidence against the null hypothesis, so we cannot reject it in favor of the alternative hypothesis.

Significance level

As before, we can set a significance level prior to running a hypothesis test if we want to have a firm cut-off for defining statistically significant results.

If the P-value , we reject the null hypothesis in favor of the alternative hypothesis.

If the P-value , we fail to reject the null hypothesis.

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