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# R calculate an empirical p value for the ratio you

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(r) Calculate an empirical p-value for the ratio you calculated in (m). What conclusion would you draw based on this p-value? Probability Result: For large samples, this sampling distribution is well modeled by an F distribution with parameters number of groups ± 1 and overall sample size ± number of groups , the degrees of freedom of the numerator and denominator respectively.

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Chance/Rossman, 2015 ISCAM III Investigation 5.4 342 Terminology Detour We will compare I group means, where each group has n i observations. The overall sample size will be denoted by N = 6 n i . H 0 : There is no treatment effect or H 0 : P 1 = … = P I H a : There is a treatment effect or H a : at least one P i differs from the rest The between-group variability will be measured by looking at the sum of the squared deviations of the group means to the overall mean, . Each group mean is weighted by the sample size of that group. We will refer to this quantity as the “sum of squares for treatment,” SST. SST = ³ ´ ¦ ± I i i i x x n 1 2 We will then “average” these valu es by considering how many groups were involved. This quantity will be referred to as the “mean square for treatments.” MST = ) 1 ( ± I SST Note, if we fix the overall mean, once we know I ± 1 of the group means, the value of the i th mean is determined. So the degrees of freedom of this quantity is I ± 1. The within-group variability will be measured by the pooled variance. In general, each term will be weighted by the sample size of that group. We will again divide by an indication of the o verall sample size across the groups. We will refer to this quantity as the “mean squares for error,” MSE. MSE = I N s n I i i i ± ± ¦ 1 2 ) 1 ( which has N ± I degrees of freedom. The test statistic is then the ratio of these “mean square” quantities: MSE MST F When the null hypothesis is true, this test statistic should be close to 1. So larger values of F provide evidence against the null hypothesis. The corresponding p-value comes from a probability distribution called the F distribution with I ± 1 and N ± I degrees of freedom. We will use this F distribution to approximate both the sampling distribution of this test statistic in repeated samples from the same population (H 0 : P 1 = = P , ´ and the randomization distribution for a randomized experiment (H 0 : no treatment effect) as long as the technical conditions (see below) are met. Because we are focusing on the variance of group means, this procedure is termed Analysis of Variance (ANOVA). With one explanatory variable, this is called one-way ANOVA. x
Chance/Rossman, 2015 ISCAM III Investigation 5.4 343 Technology Detour ANOVA In R x If you have the data in a response vector and an explanatory vector, you can use summary(aov(response~explanatory)) This will output the Mean Square values in the Mean Sq column, first for the treatment group (the disability row) and then for the error term (Residuals).

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