# Four pairs of data yield r=0.942 and the regression equation y-hat=3x. Also, y-bar=12.75. What is the best predicted value for x=4.3?

I know that the answer is 12.5, but how do you know to use the y-bar value instead of plugging the x value into the equation?

## To find the best predicted value for x=4.3, you can use the regression equation. However, since the equation only provides an estimate, the best predicted value should be closer to the mean of the y-values (y-bar). This is because the regression equation takes into account the overall trend of the data but may not always be accurate for individual predictions.

In this case, the mean of the y-values (y-bar) is given as 12.75. Since it is closer to the best estimate for x=4.3, using the y-bar value would provide a more reliable prediction. Plugging x=4.3 into the regression equation (y-hat=3x) would give you y-hat=3(4.3)=12.9, which is farther away from the mean than the y-bar value.

Therefore, it is preferable to use the y-bar value of 12.75 as the best predicted value for x=4.3.

## To find the best predicted value for x=4.3, you are correct in using the y-bar (mean of the y-values) instead of plugging the x value into the regression equation. Here's why:

The regression equation y-hat = 3x represents the line of best fit obtained from the given pairs of data. However, this equation predicts the average value of y for a given x.

When we calculate the regression equation, we are trying to minimize the overall vertical distance between the predicted y-values (y-hat) and the actual y-values. This means that the line of best fit is intended to represent the average trend or general relationship between the x and y values.

On the other hand, the y-bar (mean of the y-values) represents the average value of the y-values observed in the given data.

Using the y-bar value instead of plugging the x value into the equation is based on the assumption that the given data points are representative of the overall population. Therefore, by using the y-bar value, you are predicting the average y-value for x=4.3 based on the general trend of the data.

In summary, using the y-bar value instead of plugging the x value into the equation is appropriate when you want to estimate the overall average value of y for a given x, based on the given data and assuming the data is representative of the population.