Statistics: Overview
Statistics: Principles of Testing
Hypothesis testing of problems with one variable requires carrying out the following steps:
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State the null hypothesis and the alternative hypothesis.
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Decide on a significance level for the test.
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Compute the value of a test statistic.
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Compare the test statistic to a critical value from the appropriate probability distribution corresponding to your chosen level of significance and observe whether the test statistic falls within the region of acceptance or the region of rejection.
The general formula for computing a test statistic for making an inference about a single population is
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where observed sample statistic is the statistic of interest from the sample (usually the mean), tested value is the hypothesized population parameter (again, usually the mean), and standard error is the standard deviation of the sampling distribution divided by the positive square root of n.
The general formula for computing a test statistic for making an inference about a difference between two populations is
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where statistic1 and statistic2 are the statistics from the two samples (usually the means) to be compared, tested value is the hypothesized difference between the two population parameters (0 if testing for equal values), and standard error is the standard error of the sampling distribution, whose formula varies according to the type of problem.
The general formula for computing a confidence interval is
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where a and b are the lower and upper limits of the confidence interval, x is the point estimate (usually the sample mean), test statistic is the critical value from the table of the appropriate probability distribution (upper or positive value if z) corresponding to half of the desired alpha level, and standard error is the standard error of the sampling distribution.
Why must the alpha level be halved before looking up the critical value when computing a confidence interval? Because the rejection region is split between both tails of the distribution, as in a two-tailed test. For a confidence interval at α = .05, you would look up the critical value corresponding to an upper-tailed probability of .025.
