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In a pilot study of only 45 students a researcher found a Multiple $\mathrm{R}^{2}$ value of $.16$ (this is the squared correlation, $\mathrm{r}^{2}$, between the DV and the predicted DV) when analyzing the pilot study data with multiple regression. The study data included a dependent variable (student mathematics test scores) and two independent variables ([a] number of homework problems successfully completed the week before the test and [b] the number of minutes per day spent in mathematics instructional activities). Now ready to move beyond the pilot testing phase, using information learned from the pilot study, what size sample should the teacher seek to detect an effect of this size (Multiple $\mathrm{R}^{2}=.16$ ) with two independent variables?
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(a) alpha = was not specified in the scenario, so I will use both $.01$ and $.05$ in this example response;

(b) number of predictors $=2$ ( number of homework problems completed successfully and time in minutes in instruction on mathematics)

(d) power = was not specified, so I will use $.80$ and $.90$

(c) The effect size: the value $\mathrm{R}^{2}=.16$ must be converted to $\mathrm{f}^{2}$. In regression, $\mathrm{R}^{2}$ is just the squared Pearson correlation, $\mathrm{r}^{2}$, between the dependent variable and the predicted scores on the DV. The value $\mathrm{f}^{2}$ can be converted from $\mathrm{r}^{2}$ using the conversion file below:
by Diamond (89,036 points)

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