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Hypothesis Testing - a statistical method that uses sample data to…
Hypothesis Testing - a statistical method that uses sample data to evaluate a hypothesis about a population.
First Step: we state a hypothesis about a population. Usually the hypothesis concerns the value of a population parameter
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Alternative Hypothesis (scientific or hypothesis) 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 treatment (independent variable) has an effect on the dependent variable.
Second Step: before we select a sample, we use the hypothesis to describe what values we should expect for the sample mean, if the hypothesis is really true.
We determine exactly which sample means are consistent with the null hypothesis and which sample means are at odds with the null hypothesis.
Sample means that are likely to be obtained if null hypothesis is true; that is, sample means that are close to the null hypothesis.
Sample means that are very unlikely to be obtained if null hypothesis is true; that is, sample means that are very different from the null hypothesis.
Alpha Level (Level of Significance) - a probability value that is used to define the concept of “very unlikely” in a hypothesis test.
Critical Region - 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.
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Fourth Step: we compare the obtained sample data with the prediction that was made from the hypothesis. If the sample mean is consistent with the prediction, we conclude that the hypothesis is reasonable. But if there is a big discrepancy between the data and the prediction, we decide that the hypothesis is probably wrong.
The researcher uses the z-score value obtained in Step 3 to make a decision about the null hypothesis according to the criteria established in Step 2.
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 null hypothesis and our decision is to reject the null hypothesis. Remember, the null hypothesis states that there is no treatment effect. By rejecting null hypothesis we are concluding there is evidence that the treatment had an effect.
We do not state there is an effect. Instead, we state there is evidence for an effect. The distinction is subtle but important. By rejecting the null hypothesis, we are not proving the existence of a treatment effect (that is, we are not proving the alternative hypothesis to be true)
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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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. Adding (or subtracting) a constant changes the mean but does not change the shape of the population, nor does it change the standard deviation. 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.
Instead, we are asking what would happen if the treatment were administered to the entire population. The research study involves selecting a sample from the original population, administering the treatment to the sample, and then recording scores for the individuals in the treated sample.
Hypothesis testing is an inferential process, which means that it uses limited information as the basis for reaching a general conclusion.
2 Types of Error
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.
Type I error occurs when a researcher unknowingly obtains an extreme, nonrepresentative sample.
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.
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.
A Type II error occurs when the sample mean is not in the critical region even though the treatment has an effect on the sample.
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.
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A result is said to be significant or statistically significant if it is very unlikely to occur when the null hypothesis is true. That is, the result is sufficient to reject the null hypothesis. Thus, a treatment has a significant effect if the decision from the hypothesis test is to reject the null hypothesis.
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In a directional hypothesis test, or a one-tailed test, the statistical hypotheses ( and ) specify either an increase or a decrease in the population mean. That is, they make a statement about the direction of the effect.
Null Hypothesis - tips are not increased, the treatment does not work.
Hypothesis - tips are increased, treatment works as predicted.
A two-tailed test, on the other hand, requires a relatively large difference independent of direction.
Two-tailed tests should be used in research situations when there is no strong directional expectation or when there are two competing predictions.
Concerns
Demonstrating a statistically significant treatment effect does not necessarily indicate a substantially large treatment effect.
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.
The power of a statistical test is the probability that the test will correctly reject a false null hypothesis. That is, power is the probability that the test will identify a treatment effect if one really exists.
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