Question

Consider the sample regressions for the linear, the logarithmic, the exponential, and the log-log models. For each of the estimated models, predict y when x equals 50. Note: Round intermediate...

Answer 1

For the linear model,
`\(\hat{y}_{\text{linear}} = \text{result}\)` ; for the logarithmic model,
`\(\hat{y}_{\text{log}} = \text{result}\)` , and for the exponential and log-log models,
`\(\hat{y}_{\text{exp}} = \hat{y}_{\text{loglog}} = \text{result}\) when \(x = 50\).`

Step-by-step explanation:

In the linear model, we use the formula
`\( \hat{y}_{\text{linear}} = b_0 + b_1 \cdot x \) where \( b_0 \) and \( b_1 \)` are the intercept and slope coefficients, respectively. Substituting
`\( x = 50 \)` into this equation gives us the predicted value for y.

For the logarithmic model, the formula is
`\( \hat{y}_{\text{log}} = \log(b_0) + \log(b_1) \cdot x \)` . Similarly, we substitute
`\( x = 50 \)` into the equation to obtain the predicted value.

In the exponential model,
`\( \hat{y}_{\text{exp}} = e^(b_0 + b_1 \cdot x) \),` and for the log-log model,
`\( \hat{y}_{\text{loglog}} = e^(\log(b_0) + \log(b_1) \cdot x) \)` . In both cases, we substitute
`\( x = 50 \)` into the equations to calculate the predicted values.

These predictions provide estimates of y based on the fitted models, allowing us to understand the expected values of y at the given x value in each case.