11-08 math

# E its sampling distribution according to the null

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Unformatted text preview: ted value of MStreatment is σ2. Just as with MSregression in hypothesis testing with regression, MStreatment has a likelihood function with ! shape of a chi ­square distribution (i.e., a distribution we know exactly), but the multiplied by σ2. Therefore, we need to divide by an estimate of σ2 to get a test statistic we can use. This estimate comes from MSresidual, which is the mean square obtained from the residual sum of squares. MSresidual = ! SSresidual df residual (6) The expected value of MSresidual (according to either H0 or H1) is σ2. In fact, the sampling distribution for MSresidual also has a chi ­square shape, again multiplied by σ2. Therefore, when we calculate the ratio of MStreatment and MSresidual, σ2 cancels out, and we end up with a test statistic that doesn’t depend on any unknown population parameters. F= MStreatment MSresidual (7) F tells us how much the group means differ relative to how much they should differ just by chance. If F is large, then the group differences are larger than can be explained by chance. ! To know how large F must be to reject the null hypothesis, we need to know its likelihood function (i.e., its sampling distribution according to the null hypothesis). The distribution for F is called an F distribution. An F distribution arises any time you take the ratio of two chi ­square variables, such as MStreatment and MSresidual. Because the distribution of any chi ­ square variable depends on its degrees of freedom, the F distribution depends on the degrees of freedom for both chi ­square variables. In other words, to specify an F distribution, we must specify two df values: the degrees of freedom for the numerator and the degrees of freedom for the denominator. In this case, these are dftreatment and dfresidual. Once we know...
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## This document was uploaded on 02/25/2014 for the course PSYC 3101 at Colorado.

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