Question

Lm = Linear regression model: SALES sim 1 + POP Estimated Coefficients: Estimate (Intercept). POP 6.5534e+05 0.44138 SE 1.0187e+05 0.12004 tStat pValue 6.4329 5.5874e-10 3.6769 0.00028429 Number of observations: 275, Error degrees of freedom: 273 Root Mean Squared Error: 1.64e + 6 R-squared: 0.0472 Adjusted R-Squared: 0.0437 F-statistic vs. constant model: 13.5, p-value = 0.000284 fx >>

Answer 1

The linear regression model predicts sales
`(\(\text{SALES}\))` based on population
`(\(\text{POP}\))` with an intercept of
`\(6.5534 * 10^5\)` and a population coefficient of
`\(0.44138\).` Both coefficients are statistically significant
`(\(p < 0.001\))`, and the model is overall significant
`(\(F = 13.5, p = 0.000284\)),` though the low
`R-squared (\(0.0472\))` suggests a limited ability to explain variability in sales.

Explanation:

The provided information includes the coefficients and statistics for a linear regression model:

1. Linear Regression Model:

`\[\text{SALES} \approx 1 + 0.44138 * \text{POP}\]`

2. Estimated Coefficients:

• Intercept Estimate:
  `\(6.5534 * 10^5\)`
• Intercept Standard Error (SE):
  `\(1.0187 * 10^5\)`
• 
  `\(\text{POP}\) Coefficient Estimate: \(0.44138\)`
• 
  `\(\text{POP}\) Coefficient SE: \(0.12004\)`

3. T-Statistics and P-Values:

Intercept T-Statistic:
`\(6.4329\)`

Intercept P-Value:
`\(5.5874 * 10^(-10)\)`

`\(\text{POP}\) T-Statistic: \(3.6769\)`

`\(\text{POP}\) P-Value: \(0.00028429\)`

4. Model Summary:

• Number of Observations:
  `\(275\)`
• Error Degrees of Freedom:
  `\(273\)`
• Root Mean Squared Error (RMSE):
  `\(1.64 * 10^6\)`
• R-Squared
  `(\(R^2\)): \(0.0472\)`
• Adjusted R-Squared:
  `\(0.0437\)`
• F-Statistic vs. Constant Model:
  `\(13.5\)`
• F-Statistic P-Value:
  `\(0.000284\)`

The linear regression model equation is given by
`\(\text{SALES} = 6.5534 * 10^5 + 0.44138 * \text{POP}\).` The t-statistics for the intercept and
`\(\text{POP}\)` are calculated as the ratio of the coefficient estimate to its standard error. For the intercept:

`\[t_{\text{Intercept}} = (6.5534 * 10^5)/(1.0187 * 10^5) \approx 6.4329\]`

For
`\(\text{POP}\):`

`\[t_{\text{POP}} = (0.44138)/(0.12004) \approx 3.6769\]`

The associated p-values represent the probability of obtaining a t-statistic as extreme as the calculated value under the null hypothesis that the corresponding coefficient is zero. These low p-values
`(\(5.5874 * 10^(-10)\)` for the intercept and
`\(0.00028429\) for \(\text{POP}\))` suggest that both coefficients are statistically significant.

The F-statistic tests the overall significance of the model. In this case, the F-statistic is
`\(13.5\)` with a p-value of
`\(0.000284\)`, indicating that the model is statistically significant.

The R-squared value
`(\(0.0472\))` provides the proportion of variance in the dependent variable
`(\(\text{SALES}\))` explained by the independent variable
`(\(\text{POP}\)).` The adjusted R-squared adjusts for the number of predictors in the model, yielding
`\(0.0437\)`. The RMSE
`(\(1.64 * 10^6\))` represents the average prediction error.

In summary, the statistical output provides a comprehensive assessment of the model's coefficients, their significance, overall model fit, and predictive accuracy.