Chapter 8 Introduction to Hypothesis Testing

Hypothesis Testing Hypothesis testing is a statistical procedure Allows researchers to use sample data to draw inferences about the population of interest Although the details of a hypothesis test will change from one situation to another, the general process will remain constant

Hypothesis Testing (cont.) For this chapter, we have to understand z- scores, probability, and the distribution of sample means to create a new statistical procedure known as a hypothesis test.

Hypothesis Test A hypothesis test is a statistical method that uses sample data to evaluate a hypothesis about a population mean. Underlying logic State a hypothesis about a population State a hypothesis about a population Use the hypothesis to predict the characteristics that the sample should have Use the hypothesis to predict the characteristics that the sample should have Obtain a random sample from the population Obtain a random sample from the population Compare the obtained sample data with the prediction that was made from the hypothesis Compare the obtained sample data with the prediction that was made from the hypothesis

A hypothesis test is typically used in the context of a research study Once a researcher completes a research study, a hypothesis test is used to evaluate the results Details of the hypothesis test will change from one situation to another Details of the hypothesis test will change from one situation to another For now, we will focus on the most common hypothesis tests

Situation: A researcher is using one sample to examine one unknown population The purpose of the research is to determine the effect of the treatment on the individuals in the population. The goal is to determine what happens to the population after the treatment is administered.

Figure 8.1 The research situation of hypothesis testing Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning Begin with a known population (before the treatment) Assume the population forms a normal distribution Purpose is to determine the effect of the treatment on the individuals in the population What happens to the population after the treatment is administered?

The problem is to determine whether or not the treatment has an effect; The parameters are known for the population before treatment; The question is whether or not the population mean is changed by the treatment; To help answer the question, the researcher obtains a sample of individuals who have received the treatment. The basic research situation for hypothesis testing

To simplify the hypothesis-testing situation, one basic assumption is made about the effect of the treatment If the treatment has any effect, it is simply to add a constant amount to (or subtract a constant amount from) each individual’s score.

Remember a constant will not change the shape of the population, nor will it change the standard deviation The population after the treatment will also have the same shape as the original population and the same s.d.

The sample in the research study The goal of the hypothesis test is to determine whether or not the treatment has any effect on the individuals in the population Because the populations are usually too big, we use a sample. The hypothesis test will use the sample to test a hypothesis about the unknown population mean.

Because a hypothesis test is a formalized procedure that follows a standard series of operations, Researchers have a standardized method for evaluating the results of their research studies; Other researchers will understand how the data were evaluated and how conclusions were reached.

Hypothesis test formal structure Will use a four-step process Will be used throughout the rest of the book

Example 8.1 Psychologists note that stimulation during infancy can have profound effects on the development of infant rats. Based on data, one might theorize that increased stimulation early in life can be beneficial. Could this theory be applied to infants?

Mean weight of 2-year olds is  = 26 lbs. With a  = 4 lbs n=16 Sample parents given instructions for working with their infants At age 2, will weigh the children

We do not know what will happen to the mean weight for 2-year old children Do have a sample of 16 infants that we can be sure about. Can use this sample to draw inferences about the unknown population Follow the four steps

Steps 1. State the hypothesis 2. Set the criteria for a decision 3. Collect data and compute sample statistics 4. Make a decision

Four Steps Step 1 State the hypotheses Actually state two hypotheses Actually state two hypotheses Both in terms of population parameters Both in terms of population parameters Null hypotheses States that the treatment has no effect. States that the treatment has no effect. Identified by the symbol H o Identified by the symbol H o H stands for hypothesis O indicates that this is the zero-effect H o =  infants handled = 26 pounds

Step 1 The second hypothesis is the opposite of the null hypothesis Called the scientific or alternative hypothesis (H 1 ) States that the treatment has an effect on the dependent variable H 1 =  infants handled > < = 26 pounds

An alternative hypothesis simply states that there will be some type of change It might be necessary to specify the direction of the effect in H 1  > 26 pounds  > 26 pounds This is called directional hypothesis test Note that both hypotheses refer to a population whose mean is unknown The population of infants who receive extra handling early in life The population of infants who receive extra handling early in life

Step 2 Set the Criteria for a Decision Will eventually use the data from the sample to evaluate the credibility of the null hypothesis Will use the null hypothesis to predict the kind of sample mean that ought to be obtained We will determine exactly what sample means are consistent with the null hypothesis and what sample means are at odds with the null hypothesis

Begin by examining all the possible sample means that could be obtained in the null hypothesis is true Distribution of sample means should be centered at  = 26 The distribution of sample means is then divided into two sections.

1. Sample means that are likely to be obtained if Ho is true Those close to the null hypothesis Those close to the null hypothesis 2. Sample means that are very unlikely to be obtained if Ho is true Those that are very different from the null hypothesis Those that are very different from the null hypothesis

The High probability samples are located in the center of the distribution and have sample means close to the value specified in the null hypothesis. The low-probability samples are located in the extreme tails of the distribution. After the distribution has been divided in this way, we can compare our sample data with the values in the distribution We can determine whether our sample mean is consistent with the null hypothesis

Figure 8.2 The set of potential samples is divided into those that are likely to be obtained and those that are very unlikely if the null hypothesis is true.

Figure 8.2 The distribution of sample means if the null hypothesis is true Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

Alpha Level To find the boundaries that separate the high-probability samples from the low- probability samples, we must define exactly what is meant by “low” probability and “high” probability. This is accomplished by selecting a specific probability value, which is known as the level of significance or the alpha level This is accomplished by selecting a specific probability value, which is known as the level of significance or the alpha level for the hypothesis test.

The alpha (  ) value is a small probability that is used to identify the low-probability samples.       With a  we will separate the most likely 95% of the sample means (the central values)

Extreme Values The extremely unlikely values, as defined by the alpha level, make up what is called the critical region Extreme values are inconsistent with the null hypothesis If data produce a sample mean that is located in the critical region, we will conclude that the data are inconsistent with the null hypothesis

Technically, the critical region is defined by sample outcomes that are very unlikely to occur if the treatment has no effect That is, if the null hypothesis is true It is almost impossible if there is no treatment effect

The boundaries for the critical region To determine the exact location for the boundaries that define the critical region Use the alpha-level probability Use the alpha-level probability Unit normal table Unit normal table

 Find the boundaries that separate the extreme 5% from the middle 95% Split the 5% 2.5% (or ) in each tail 2.5% (or ) in each tail Z = +/ Thus, for any normal distribution, the extreme 5% is in the tails of the distribution beyond z = 1.96 and z = The values define the boundaries of the critical region for a hypothesis test using 

Figure 8.3 The critical region for an alpha of.05 Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

 1% or.0100 is split between the two tails The proportion in each tail is.0050 z= +/- 2.58

.01% or.0010 is split between the two tails The proportion in each tail is.0005 z= +/- 3.30

Collect Data and Compute Sample Statistics Step 3 Collect sample data Collect the data after the sample has been selected Assures an honest objective evaluation of data Assures an honest objective evaluation of data Raw data are summarized with the appropriate statistics Compute the sample mean (in this example) Compute the sample mean (in this example) Compare the sample mean with the null hypothesis Compare the sample mean with the null hypothesis

To compare the sample mean with the null hypothesis, compute a z-score that describes exactly where the sample mean is located relative to the hypothesized population mean from H o Z = M –   M  M M = sample mean z = sample mean – hypothesized population mean  Standard error between M and 

Make a Decision Use the z-score value obtained in Step 3 to make a decision about the null hypothesis according to the criteria established in Step 2 Two possible decisions Accept the null hypothesis Accept the null hypothesis Reject the null hypothesis Reject the null hypothesis Sample data fall into critical region

Rejecting the Null Hypothesis vs. Proving the Alternative Hypothesis The reason for focusing on the null hypothesis as compared to the alternative hypothesis comes from the limitations of inferential logic Remember that we want to use the sample data to draw conclusions, or inferences, about a population

Logically, it is easier to demonstrate that a universal (population) hypothesis is false than to demonstrate that it is true It would be difficult to state “the treatment has an effect” as the hypothesis and then try to prove that this is true Therefore, we state the null hypothesis “the treatment has no effect” and try to show that it is false

In the end, we still demonstrate that the treatment does have an effect. We find support for the alternative hypothesis by disproving (rejecting) the null hypothesis