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Chp. 8 - Introduction to Hypothesis Testing - Coggle Diagram

- - - - 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.
        
        Thus, we assume that the population after treatment has the same shape as the original population and the same standard deviation as the original population.
    - - Note that the unknown population is actually hypothetical (the treatment is never administered to the entire population). Instead, we are asking what would happen if the treatment were administered to the entire population.
        
        A hypothesis test is a formalized procedure that follows a standard series of operations. Other researchers will recognize and understand exactly how the data was evaluated and how conclusions were reached.
  - - - Null hypothesis: states that the treatment has no effect (there is no change, no effect, no difference) on the general population.
        
        Scientific, or alternative, hypothesis: states that the treatment has an effect (a change, a difference, or a relationships) on the dependent variable for the general population.
        
        In an experiment, H1 predicts that the independent variable (treatment) does have an effect on the dependent variable.
        
        The null hypothesis and the alternative hypothesis are mutually exclusive and exhaustive. They cannot both be true. The data will determine whether to reject or fail to reject the null hypothesis.
        
        In an experiment, H0 predicts that the independent variable (treatment) has no effect on the dependent variable (scores) for the population.
      - Step 2: Set the Criteria for a Decision
        
        The data will either be consistent with the null hypothesis or tend to refute the null hypothesis. If there is a big discrepancy between the data and the hypothesis, we will conclude that the hypothesis is wrong.
        
        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 (as defined by the alpha level) 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.
        
        To determine the exact location for the boundaries that define the critical region, we use the alpha-level probability and the unit normal table.
        
        Step 3: Collect Data & Compute Sample Statistics
        
        Data is collected after the researcher has stated the hypotheses and established the criteria for a decision. This ensures that a researcher makes an honest, objective evaluation of the data and does not tamper with the decision criteria after the experimental outcome is known.
        
        Next, the raw data from the sample are summarized with the appropriate statistics. Now it is possible for the researcher to compare the sample mean (from the data) with the null hypothesis. This is important: comparing the data with the hypothesis.
        
        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.
        
        Step 4: Make a Decision
        
        The researcher uses the z-score to make a decision about the null hypothesis according to the criteria established.
        
        Outcome 1: The sample data are located in the critical region. By definition, a sample value in the critical region is very unlikely to occur if the null hypothesis is true. Therefore, we conclude that the sample is not consistent with and our decision is to reject the null hypothesis. Remember, the null hypothesis states that there is no treatment effect. By rejecting we are concluding there is evidence that the treatment had an effect.
        
        Outcome 2: The sample data are not in the critical region. In this case, the sample mean is reasonably close to the population mean specified in the null hypothesis (in the center of the distribution). Because the data do not provide strong evidence that the null hypothesis is wrong, our conclusion is to fail to reject the null hypothesis. This conclusion means that there is no evidence for a treatment effect.
        
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  - - - z-Score formula: z = (sample mean - hypothesized population mean) / (standard error between M and μ)
- - - - Remember: samples are not expected to be identical to their populations, and some extreme samples can be very different from the populations they are supposed to represent. If a researcher selects one of these extreme samples by chance, then the data from the sample may give the appearance of a strong treatment effect, even though there is no real effect.
        
        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 the researcher concludes that there is evidence for a treatment effect when in fact the treatment has no effect.
        
        The alpha level for a hypothesis test is the probability that the test will lead to a Type I error. That is, the alpha level determines the probability of obtaining sample data in the critical region even though the null hypothesis is true.
        
        In summary, whenever the sample data are in the critical region, the appropriate decision for a hypothesis test is to reject the null hypothesis. Normally this is the correct decision because the treatment has caused the sample to be different from the original population.
  - - - A Type II error occurs when a researcher fails to reject a null hypothesis that is in fact false. In a typical research situation, a Type II error means that the hypothesis test has failed to detect a real treatment effect.
        
        The consequences of a Type II error are usually not as serious as those of a Type I error.
        
        Unlike a Type I error, it is impossible to determine a single, exact probability for a Type II error. Instead, the probability of a Type II error depends on a variety of factors (such as sample size and effect size) and therefore is a function of several factors, rather than a specific number, like alpha, that the researcher selects.
        
        The probability of a Type II error is represented by the symbol β , the Greek letter beta .
  - - - The largest permissible value is α = .05
        
        Because the consequences of a Type I error can be relatively serious, many researchers and scientific publications prefer to use a more conservative alpha level such as .01 or .001 to reduce the risk that a false report is published and becomes part of the scientific literature.
  - - - Type II error (beta): failing to reject a false null hypothesis. The sample data are not in the critical region even though the treatment has an effect on the sample.
- - - - Step 2: Locate the critical region.
        
        Step 3: Compute the test statistic (the z-score).
        
        Step 4: Make a decision about the null hypothesis.
  - - - The number of scores in the sample: decreasing the number of scores in the sample produces a larger standard error and a smaller value (closer to zero) for the z-score. If all other factors are held constant, a larger sample is more likely to result in a significant treatment effect.
  - - - Independent observations: two events (or observations) are independent if the occurrence of the first event has no effect on the probability of the second event.
        
        The Value of σ Is Unchanged by the Treatment: we assume that the standard deviation for the unknown population (after treatment) is the same as it was for the population before treatment.
        
        Normal sampling distribution: To evaluate hypotheses with z-scores, we have used the unit normal table to identify the critical region. This table can be used only if the distribution of sample means is normal.
- - - - A one-tailed test allows you to reject the null hypothesis when the difference between the sample and the population is relatively small, provided the difference is in the specified direction.
        
        A two-tailed test, requires a relatively large difference independent of direction.
- - - - As we present different hypothesis tests, we will also present different options for measuring and reporting effect size . The goal is to measure and describe the absolute size of the treatment effect in a way that is not influenced by the number of scores in the sample.
        
        A measure of effect size is intended to provide a measurement of the absolute magnitude of a treatment effect, independent of the size of the sample(s) being used.
- - - - Thus, researchers may calculate the power of a hypothesis test before they actually conduct the experiment. In this way, they can determine the probability that the results will be significant (reject ) if the treatment really produces an effect before investing time and resources to conduct the study.
        
        Factors such as the sample size, the size of the treatment effect, and the value chosen for the alpha level can all influence a hypothesis test as well as its power.
  - - - Before a study is conducted, researchers can compute power using different sample sizes for any given assumption about effect size to determine the probability that their research will successfully reject the null hypothesis.
  - - - As the magnitude of measures of effect size, such as Cohen’s d, increase, so does statistical power.