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Statistical significance: hypothesis testing regarding one mean

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In the context of hypothesis testing regarding one mean, the test (Z or t) may be statistically significant at the 5% level, but not significant in a practical sense at all. Illustrate the meaning of this statement by citing 3 examples from any area of interest.

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In a 776 word solution, the response explains the issues in good detail complete with examples.

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When an independent variable appears to have an effect, it is very important to be able to state with confidence that the effect was really due to the variable and not just due to chance. For instance, consider a hypothetical experiment on a new antidepressant drug. Ten people suffering from depression were sampled and treated with the new drug (the experimental group); an additional 10 people were sampled from the same population and were treated only with a placebo (the control group). After 12 weeks, the level of depression in all subjects was measured and it was found that the mean level of depression (on a 10-point scale with higher numbers indicating more depression) was 4 for the experimental group and 6 for the control group. The most basic question that can be asked here is: "How can one be sure that the drug treatment rather than chance occurrences were responsible for the difference between the groups?" It could be that by chance, the people who were randomly assigned to the treatment group were initially somewhat less depressed than those randomly assigned to the control group. Or, it could be that, by chance, more pleasant things happened to the ...

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