Minsu is an engineer who ran a simple linear regression on a sample size of 33 to
explain the variation in battery life against ambient temperature. The sample
standard deviation of battery life was 1.625 hours, and the explained variation was
30.2.
The coefficient of determination for the model is closest to:
A. 0.357
B. 0.556
C. 0.643
The coefficient of determination, denoted as R-squared, is the proportion of the total variation in the response variable that is explained by the regression model. It ranges between 0 and 1, where 1 indicates that the model explains all the variation, and 0 indicates that the model does not explain any variation.
The formula for R-squared in simple linear regression is:
R-squared = explained variation / total variation
where the total variation is the sum of the explained variation and the unexplained variation.
We can find the unexplained variation by taking the square root of the sample variance of the residuals, which is:
sqrt(sample variance of residuals) = sqrt((1/n-2) * sum(residuals^2))
where n is the sample size and residuals are the differences between the actual values and the predicted values from the regression model.
Since we are not given the values of the residuals, we cannot compute the unexplained variation directly. However, we can use the sample standard deviation of battery life as an estimate of the standard deviation of the residuals, which is a common practice in statistics.
The standard deviation of the residuals is related to the sample standard deviation of battery life by the formula:
standard deviation of residuals = sample standard deviation of battery life / sqrt(1 – R-squared)
Rearranging this formula, we get:
R-squared = 1 – (sample standard deviation of residuals / sample standard deviation of battery life)^2
Substituting the given values, we get:
R-squared = 1 – (sqrt(1/n-2 * sum(residuals^2)) / 1.625)^2
Since we do not have the values of the residuals, we cannot compute the sum of their squares, which is needed to compute R-squared. However, we can use the explained variation as an estimate of the total variation, which is also a common practice in statistics. The total variation is the sum of the explained variation and the unexplained variation, so we have:
total variation = explained variation / R-squared
Substituting the given values, we get:
total variation = 30.2 / R-squared
Using this equation and the formula for R-squared, we can set up a system of equations:
total variation = 30.2 / R-squared total variation = sample variance of battery life = 1.625^2
Solving for R-squared, we get:
R-squared = 1 – (sqrt(total variation) / sample standard deviation of battery life)^2
Substituting the values, we get:
R-squared = 1 – (sqrt(1.625^2) / sample standard deviation of battery life)^2 R-squared = 1 – (1.625 / sample standard deviation of battery life)^2 R-squared = 1 – (1.625 / 1.625)^2 R-squared = 1 – 1 R-squared = 0
Therefore, the coefficient of determination for the model is 0, which means that the model does not explain any variation in the response variable. This result is surprising, given that the explained variation is 30.2. It suggests that there is a problem with the analysis or the data.