A researcher evaluating the significance of a regression equation obtains an F-ratio with df = 1, 23. How many pairs of scores, X and Y values, are in the sample? Explain what you can as it applies.
To determine the number of pairs of scores (X and Y values) in the sample based on the given F-ratio with df = 1, 23, we need to understand the relationship between sample size and degrees of freedom.
In regression analysis, the F-ratio is calculated by taking the ratio of two mean square values: the mean square due to regression (MSR) and the mean square error (MSE). The formula for the F-ratio is:
F = MSR / MSE
The degrees of freedom associated with the F-ratio are related to the sample size and the number of predictor variables in the regression equation.
For the MSR, the degrees of freedom can be expressed as (number of predictors, dfR) since it measures the variability explained by the regression equation. In this case, dfR = 1 because there is only one predictor.
For the MSE, the degrees of freedom can be expressed as (sample size - number of predictors - 1, dfE) since it measures the variability not explained by the regression equation. In this case, dfE = 23.
Given that the F-ratio's degrees of freedom are df = 1, 23, we can deduce that the sample size is equal to:
Sample size = dfE + dfR + 1
= 23 + 1 + 1
= 25
Therefore, there are 25 pairs of scores (X and Y values) in the sample.