Cheatography

# STATISTICAL ANALYSIS - Hypothesis Testing Cheat Sheet by vxnxssx

Intro to Hypothesis Testing

### HYPOTHESIS TESTING

 A technique to help determine whether a specific treatment has an effect on the indivi­duals in a popula­tion. Goal To rule out chance (sampling error) as a plausible explan­ation for the results from a research study. Usage: To evaluate the results from a research study in which: 1. A sample is selected from the popula­tion.​ 2. The treatment is admini­stered to the sample.​ 3. After treatment, the indivi­duals in the sample are measured. ​

### HYPOTHESIS TESTING

If the indivi­duals in the sample are noticeably different from the indivi­duals in the original popula­tion, we have evidence that the treatment has an effect. However, it is also possible that the difference between the sample and the population is simply sampling error ​

### HYPOTHESIS TESTING

 Purpose: 1. The difference between the sample and the population can be explained by sampling error (there does not appear to be a treatment effect)​ 2. The difference between the sample and the population is too large to be explained by sampling error (there does appear to be a treatment effect).​

### ERRORS

 TYPE I ERRORS Occurs when the sample data appear to show a treatment effect when, in fact, there is none. ​In this case the researcher will reject the null hypothesis and falsely conclude that the treatment has an effect. ​ These are caused by unusual, unrepr­ese­ntative samples. Just by chance the researcher selects an extreme sample with the result that the sample falls in the critical region even though the treatment has no effect. ​ The hypothesis test is structured so that Type I errors are very unlikely; specif­ically, the probab­ility of a Type I error is equal to the alpha level.​

### ERRORS

 TYPE II Occurs when the sample does not appear to have been affected by the treatment when, in fact, the treatment does have an effect. ​ In this case, the researcher will fail to reject the null hypothesis and falsely conclude that the treatment does not have an effect. ​ Type II errors are commonly the result of a very small treatment effect. Although the treatment does have an effect, it is not large enough to show up in the research study. ​

Actual Solution

### MEASURING EFFECT SIZE

 A hypothesis test evaluates the statis­tical signif­icance of the results from a research study. ​ That is, the test determines whether or not it is likely that the obtained sample mean occurred without any contri­bution from a treatment effect. The hypothesis test is influenced not only by the size of the treatment effect but also by the size of the sample. ​ Thus, even a very small effect can be signif­icant if it is observed in a very large sample. Because a signif­icant effect does not necess­arily mean a large effect, it is recomm­ended that the hypothesis test be accomp­anied by a measure of the effect size. We use Cohen=s d as a standa­rdized measure of effect size. Much like a z-score, Cohen=s d measures the size of the mean difference in terms of the standard deviat­ion.​

### NULL HYPOTHESIS

 Step 1 State the hypotheses and select an α level. The null hypothesis, H0, always states that the treatment has no effect (no change, no differ­ence). According to the null hypoth­esis, the population mean after treatment is the same is it was before treatment. The α level establ­ishes a criterion, or "­cut­-of­f", for making a decision about the null hypoth­esis. The alpha level also determines the risk of a Type I error.

### CRITICAL REGION

 Step 2 Locate the critical region. The critical region consists of outcomes that are very unlikely to occur if the null hypothesis is true. That is, the critical region is defined by sample means that are almost impossible to obtain if the treatment has no effect. The phrase “almost imposs­ible” means that these samples have a probab­ility (p) that is less than the alpha level.​

### TEST STATISTIC

 Step 3 Compute the test statistic. The test statistic (in this chapter a z-score) forms a ratio comparing the obtained difference between the sample mean and the hypoth­esized population mean versus the amount of difference we would expect without any treatment effect (the standard error). ​

### ALPHA LEVEL

 Step 4 A large value for the test statistic shows that the obtained mean difference is more than would be expected if there is no treatment effect. If it is large enough to be in the critical region, we conclude that the difference is signif­icant or that the treatment has a signif­icant effect. In this case we reject the null hypoth­esis. If the mean difference is relatively small, then the test statistic will have a low value. In this case, we conclude that the evidence from the sample is not suffic­ient, and the decision is fail to reject the null hypoth­esis. ​

### DIRECT­IONAL TESTS

 Includes the direct­ional prediction in the statement of the hypotheses and in the location of the critical region. ​ For example, if the original population has a mean of μ = 80 and the treatment is predicted to increase the scores, then the null hypothesis would state that after treatm­ent:​ H0: μ < 80 (there is no increase) ​ In this case, the entire critical region would be located in the right-hand tail of the distri­bution because large values for M would demons­trate that there is an increase and would tend to reject the null hypoth­esis.​

### POWER OF A HYPOTHESIS TEST

 The power of a hypothesis test is defined is the probab­ility that the test will reject the null hypothesis when the treatment does have an effect. ​ The power of a test depends on a variety of factors including the size of the treatment effect and the size of the sample. ​