Question

QUESTION 5 In a study of reaction times, the time to respond to a visual stimulus () and the time to respond to an auditory stimulus () were recorded for each of subjects Times were measured in thousandths of a second The results are presented in the following table Visual 208 177 250 248 152 164 196 238 Auditory 200 175 233 234 155 161 190 230 Compute the least squares regression line for predicting audatory response time from visual response time). Following are computed based on the data x=1633, Zy=1578, x2 = 343657, 2y =318616, and xy =330801 D - 0.842152 + 25 345634x = 25.345634x - 0.842152x D - 25 345634-0.842152

Answer 1

The least square estimates of the regression coefficients $\widehat{\beta_0}\,,\widehat{\beta_1}$ of the regression equation $Y=\beta_0+\beta_1X+\epsilon$ is given by,

$\widehat{\beta_1}=\frac{\sum xy-\frac{\sum y\sum x}{n}}{\sum x^2-\frac{(\sum x)^2}{n}}$

$\widehat{\beta_0}=\frac{\sum y}{n}-\widehat{\beta_1}\left ( \frac{\sum x}{n} \right)$

Substituting the values in the given equations for n=8 we get,

$\widehat{\beta_1}=\frac{330801-\frac{1578\times 1633}{8}}{343657-\frac{(1633)^2}{8}}=0.84215$

$\widehat{\beta_0}=\frac{1578}{8}-0.8421\times \frac{1633}{8}=25.3456$

Therefore the predicted regression line is $\widehat{Y}=\widehat{\beta_0}+\widehat{\beta_1}X\Rightarrow \widehat{Y}=25.3456+0.84215X$ . Hence the last otion is correct.