8.1 The logic of hypothesis testing

Hypothesis testing is a statistical procedure that allows researchers to use sample data to draw inferences about the population of interest.

Hypothesis testing procedure:
1. State a hypothesis about a population. Usually the hypothesis concerns the value of a population parameter.
2. Use the hypothesis to predict the characteristics that the sample should have.
3. Obtain a random sample from the population.
4. Compare the obtained sample data with the prediction that made from the hypothesis.
a. If the sample mean is consistent with the prediction, then we conclude that the hypothesis is reasonable.
b. If there is a big discrepancy between the data and the prediction, then we decide that the hypothesis is wrong.

8.1b Step1: State the Hypothesis

Null hypothesis (H0) states that in the general population there is no change, no difference, or no relationship. In the context of an experiment, H0 predicts that the independent variable (treatment) has no effect on the dependent variable (scores) for the population.

Alternative hypothesis (H1) – opposite of H0 – states that there is a change, a difference or a relationship for the general population. In the context of an experiment, H1, predicts that the independent variable (treatment) does have an effect on the dependent variable.

8.1c Step 2: Set the Criteria for a Decision

We determine exactly which sample means are consistent with the null hypothesis and which sample means are at odds with the null hypothesis.

The alpha level, or the level of significance, is a probability value that is used to define the concept of “very unlikely” in a hypothesis test.

The critical region is composed of the extreme sample values that are very unlikely to be obtained if the null hypothesis is true. The boundaries for the critical region are determined by the alpha level. If sample data fall in the critical region, the null hypothesis is rejected.

A larger alpha means that the boundaries for the critical region move closer to the center of the distribution.

8.1d Step 3: Collect Data and Compute Sample Statistics
The comparison is accomplished by computing a z-score that describes exactly where the sample mean is located relative to the hypothesized population mean from H0.
Om(Standard Error) = o/sqrtn
Z = sample mean – hypothesized population mean / standard error between M and u.

8.1e Step4: Make a Decision
There are two possible outcomes: to reject the null hypothesis or not to reject.

A z – score near zero indicates that the data support the null hypothesis.
A z – score value in the critical region means that the sample is not consistent with the null hypothesis.

8.2 Uncertainty and Errors in Hypothesis Testing
Hypothesis testing is an inferential process, which means that is uses limited information as the basis for reaching a general conclusion.

8.2a Type I Errors
A Type I error occurs when a researcher rejects a null hypothesis that is actually true. In a typical research situation, a Type I error means that the researcher concludes that a treatment does have an effect when, in fact, it has no effect.

With an alpha level of equal 0.05, only 5% of the samples have means in the critical region. Therefore, there is only a 5% probability (p = 0.05) that one of these samples will be obtained. The alpha level determines the probability of a Type I error.

8.2b Type II Errors
A Type II error occurs when a researcher fails to reject a null hypothesis that is really false. In a typical research situation, a Type II error means that the hypothesis test has failed to detect a real treatment effect.

A Type II error occurs when the sample mean is not in the critical region even though the treatment has had an effect on the sample. This happens when the effect of the treatment is relatively small.

A hypothesis test always leads to one of two decisions:
1. The