91Ó°ÊÓ

Consider the computer output below. The regression equation is \(Y=12.9+2.34 x\) $$\begin{array}{lrrll}\text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\\\\text { Constant } & 12.857 & 1.032 & ? & ? \\\\\text { X } & 2.3445 & 0.1150 & ? & \text { ? }\end{array}$$ \(\begin{array}{ll}\mathrm{S}=1.48111 & \mathrm{R}-\mathrm{Sq}=98.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj})=97.9 \%\end{array}\) Analysis of Variance $$\begin{array}{lrrrl}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } \\\\\text { Regression } & 1 & 912.43 & 912.43 & ? \\\\\text { Residual Error } & 8 & 17.55 & ? & \\\\\text { Total } & 9 & 929.98 & &\end{array}$$ (a) Fill in the missing information. You may use bounds for the \(P\) -values (b) Can you conclude that the model defines a useful linear relationship? (c) What is your estimate of \(\sigma^{2}\) ?

Step by step solution

01

Calculate the Missing T-values

The T-value for each term in a regression equation is calculated by dividing the coefficient by its standard error. For the constant term:\[ T = \frac{12.857}{1.032} \approx 12.46 \]For the \(X\) term:\[ T = \frac{2.3445}{0.1150} \approx 20.38 \]

02

Determine P-values Bounds

The P-value is determined based on the T-values and degrees of freedom. Given the high T-values computed:- The P-value for each term is less than the typical significance level \( \alpha = 0.05 \), which usually signifies statistical significance. Thus, the P-values are \(< 0.05\).

03

Calculate the Missing Mean Square Error

Mean Square Error (MSE) is determined by dividing the sum of squares of residual error by its degrees of freedom:\[ \text{MSE} = \frac{\text{SS (Residual)}}{\text{DF (Residual)}} = \frac{17.55}{8} = 2.19375 \]

04

Determine F-Statistic

The F-statistic is calculated by dividing the Mean Square of Regression (MSR) by the Mean Square Error (MSE):\[ F = \frac{\text{MS (Regression)}}{\text{MSE}} = \frac{912.43}{2.19375} \approx 415.82 \]

05

Assess the Usefulness of the Model

The high F-statistic and low P-values suggest that the regression model is statistically significant. Therefore, we can conclude that the model defines a useful linear relationship.

06

Estimate Sigma Squared \(\sigma^2\)

The estimate of \(\sigma^2\), the variance of the errors, is the mean square error calculated in Step 3: \[ \sigma^2 = 2.19375 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

T-value

In regression analysis, the T-value is crucial to determine if individual predictors are significantly contributing to the model. It is calculated by dividing a predictor's coefficient by its standard error. This ratio gives us an idea of how many standard deviations the coefficient is away from zero.

In our example, the T-value for the constant term was calculated as approximately 12.46, and for the predictor X, it was about 20.38.

• A high absolute T-value means the coefficient is much larger than the corresponding standard error, suggesting the predictor is important.
• Generally, a T-value greater than |2| indicates statistical significance.

Thus, both terms in our regression equation have significant T-values, showing strong statistical evidence to include them in the model.

P-value

The P-value assesses how strongly the data contradicts the null hypothesis, which assumes no effect or no relationship. A smaller P-value indicates stronger evidence against the null hypothesis.

In the regression model, the P-value helps us determine whether the predictor variables are statistically significant:

• If P-value < 0.05, the result is considered statistically significant, which suggests the predictor contributes meaningfully to the model.
• P-values closer to 0 indicate stronger evidence.

Using our T-values (greater than |2| for both coefficients), we deduced that their P-values are less than 0.05. This is significant, implying that both the constant and the predictor X are valuable components of the model.

Mean Square Error (MSE)

Mean Square Error (MSE) is a key metric in regression analysis that helps measure the average squared differences between the observed actual outcomes and the outcomes predicted by the model. It's a gauge of how well the model fits the data.

In the analysis:

• MSE is calculated by dividing the Residual Sum of Squares (SS) by their degrees of freedom.
• In this instance, MSE = 2.19375, as determined by \( \frac{17.55}{8} \).

A lower MSE suggests that the model's predictions are close to the actual data points, indicating a good fit. High MSE might indicate the model is not capturing the underlying patterns correctly.

F-statistic

The F-statistic in regression analysis is used to determine if the overall regression model is a better fit compared to a model with no predictors at all. It compares the model's predictive power by evaluating the significance of the regression.

Here’s how it's calculated:

• Divide the Mean Square Regression (MSR) by the Mean Square Error (MSE).
• Here, \( F = \frac{912.43}{2.19375} \approx 415.82 \).

Such a high F-statistic, as in this case, implies that the model is statistically significant. This means it well explains the variability in the data due to the predictor variables when compared to random chance alone. Consequently, we conclude that our model accurately defines a significant relationship between X and Y.