Sample Statistics Class Handout

SAMPLE MGT 249 HANDOUTS

Statistical Inference: Hypothesis Testing

Our objectives for this section are to be able to formulate statistical hypotheses, determine what data should be used to test the hypotheses, perform the appropriate test, and analyze the results. We will also discuss how to use the computer to perform the necessary calculations.
Hypothesis testing is the second major topic of statistical inference. We will use hypothesis tests throughout the rest of the quarter.

Hypothesis Testing Procedure

To structure the hypothesis testing procedure, we can follow a few steps. Details of these steps, including the necessary vocabulary, will be given in the pages that follow.
1. Formulate the null hypothesis.
2. Formulate the alternative hypothesis and determine if the test is 1 or 2 sided.
3. Select the appropriate test statistic.
4. Set the sample size, collect the data and compute the test statistic selected in 3.
5. Determine the level of error that can be tolerated (a).
6. Find the critical region or the p-value.
7. If the computed test statistic falls in the critical region (or if the p-value is less than a) reject the null hypothesis.

Steps 1 and 2: Establishing the Hypotheses

A statistical hypothesis is an assumption or statement concerning one or more populations. Most often we make and test hypotheses about the parameters of a particular distribution.
We will always formulate two hypotheses. The first is called the null hypothesis, and is denoted H0. The second is called the alternative hypothesis, and is denoted Ha.
Forming the null hypothesis:
When forming the null hypothesis, we need to ask ourselves if we have past information on the parameter or distribution in question.
1. If we have past information, the null hypothesis states that nothing has changed, that the status quo is maintained.
2. If we don't have past information, we can frequently base the null hypothesis on a claim or on what is supposed to be.
3. Frequently, we formulate the null hypothesis with the hope of rejecting it.
Example: A certain cold vaccine has been shown to be only 25% effective after a period of 2 years. A new vaccine is available, which is more expensive. It is hoped that the new vaccine will be more effective.

Ho: p = .25
Ha: p > .25
In the past 25% have been aided by the current vaccine. It is hoped that the new drug will be more effective, so that we hope to reject the null hypothesis in favor of the alternative.
As we will see in the next section, testing the hypothesis is based on probabilities of making erroneous conclusions. Hence, one other way of determining what the null hypothesis should be is that we want a small probability of rejecting it when it is actually true.
The null hypothesis should always contain the equality part (it can be =, less than or equal to, or greater than or equal to).
The form of the alternative hypothesis is important in what we develop later. If the alternative hypothesis has the form "not equal to", then the test is said to be a 2-tailed (or 2-sided) test. If the alternative hypothesis is an inequality (< or >), the test is 1 tailed.

Step 3: Selecting the Appropriate Test Statistic

Choosing the appropriate test statistic is very similar to deciding which confidence interval should be used. To begin we will concentrate on the same 3 cases we concentrated on when doing estimation:
1. We are interested in the mean of a normal distribution, and we know the population standard deviation. In this case the test statistic is
  
  .
2. We are interested in the mean of a normal distribution, but we do not know the population standard deviation. In this case the test statistic is
  
  .
3. We are interested in the proportion of successes of a binomial distribution. In this case the test statistic is .
  
  Strictly speaking, for this case to be valid we should have
The values m0 and p0 refer to the hypothesized values.

Last updated Feb. 7, 1997