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Chapter 8 - Coggle Diagram
Chapter 8
μ
is obtained from the null hypothesis. The z-score test statistic identifies the location of the sample mean in the distribution of sample means. Expressed in words, the z-score formula is
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Make a decision. If the obtained z-score is in the critical region, reject because it is very unlikely that these data would be obtained if were true. In this case, conclude that the treatment has changed the population mean. If the z-score is not in the critical region, fail to reject because the data are not significantly different from the null hypothesis. In this case, the data do not provide sufficient evidence to indicate that the treatment has had an effect.
Hypothesis testing is structured as a four-step process that is used throughout the remainder of the book.
State the hypotheses, and select an alpha level. The null hypothesis states that there is no effect or no change. In this case, states that the mean for the population after treatment is the same as the mean before treatment. The alpha level, usually
, provides a definition of the term very unlikely and determines the risk of a Type I error. Also state an alternative hypothesis , which is the exact opposite of the null hypothesis.
Locate the critical region. The critical region is defined as sample outcomes that would be very unlikely to occur if the null hypothesis is true. The alpha level defines “very unlikely.”
Collect the data and compute the test statistic. The sample mean is transformed into a z-score by the formula
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Whatever decision is reached in a hypothesis test, there is always a risk of making the incorrect decision. There are two types of errors that can be committed:
A Type I error is defined as rejecting a true . This is a serious error because it results in falsely reporting a treatment effect. The risk of a Type I error is determined by the alpha level and therefore is under the experimenter’s control.
A Type II error is defined as the failure to reject a false . In this case, the experiment fails to detect an effect that actually occurred. The probability of a Type II error cannot be specified as a single value and depends in part on the size of the treatment effect. It is identified by the symbol
In addition to using a hypothesis test to evaluate the significance of a treatment effect, it is recommended that you also measure and report the effect size. One measure of effect size is Cohen’s d, which is a standardized measure of the mean difference. Cohen’s d is computed as
The size of the sample influences the outcome of the hypothesis test, but has little or no effect on measures of effect size. As sample size increases, the likelihood of rejecting the null hypothesis also increases. The variability of the scores influences both the outcome of the hypothesis test and measures of effect size. Increased variability reduces the likelihood of rejecting the null hypothesis and reduces measures of effect size
When a researcher expects that a treatment will change scores in a particular direction (increase or decrease), it is possible to do a directional, or one-tailed, test. The first step in this procedure is to incorporate the directional prediction into the hypotheses. To locate the critical region, you must determine what kind of data would refute the null hypothesis by demonstrating that the treatment worked as predicted. These outcomes will be located entirely in one tail of the distribution.
As the size of the treatment effect increases, statistical power increases. Also, power is influenced by several factors that can be controlled by the experimenter:
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The power of a hypothesis test is defined as the probability that the test will correctly reject the null hypothesis.
To determine the power for a hypothesis test, you must first identify the boundaries for the critical region. Then, you must specify the magnitude of the treatment effect, the size of the sample, and the alpha level. With these assumptions, the power of the hypothesis test is the probability of obtaining a sample mean in the critical region.
One of the most important concepts in this chapter is standard error. The standard error tells how much error to expect if you are using a sample mean to represent a population mean.
The location of each M in the distribution of sample means can be specified by a z-score:
One of the most important concepts in this chapter is standard error. The standard error tells how much error to expect if you are using a sample mean to represent a population mean.
The location of each M in the distribution of sample means can be specified by a z-score: