Math Problem Statement

X2X4 SUMMARY OUTPUT

Regression Statistics

Multiple R

0.93366673

R Square

0.87173357

Adjusted R Square

0.84608028

Standard Error

1.37780377

Observations

19

ANOVA

df

SS

MS

F

Significance F

Regression

3

193.524852

64.5082838

33.9813595

6.245E-07

Residual

15

28.4751485

1.89834323

Total

18

222

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

0.23845632

0.7139416

0.33399976

0.74300357

-1.2832742

1.76018683

-1.2832742

1.76018683

9

0.26259441

0.11145711

2.35601314

0.03249596

0.02502921

0.5001596

0.02502921

0.5001596

4

0.17887402

0.20841216

0.85827055

0.40424941

-0.265346

0.62309403

-0.265346

0.62309403

1156

-0.0014421

0.00026162

-5.5122271

5.9671E-05

-0.0019998

-0.0008845

-0.0019998

-0.0008845

Analyze the regression output and interpret the results, including Regression Statistics, ANOVA table, and Coefficients significance.

Solution

This regression output provides key details about the regression model's fit, significance, and the individual coefficients. Here's a breakdown of the results:

1. Regression Statistics:

  • Multiple R: 0.9337, indicating a strong correlation between the observed and predicted values.
  • R Square: 0.8717, meaning about 87.17% of the variance in the dependent variable is explained by the model.
  • Adjusted R Square: 0.8461, adjusting for the number of predictors, which is slightly lower than R Square due to this adjustment.
  • Standard Error: 1.3778, representing the average distance that the observed values fall from the regression line.
  • Observations: 19, showing the sample size.

2. ANOVA Table:

  • Regression (df = 3): This has an SS (Sum of Squares) of 193.5249, with an MS (Mean Square) of 64.5083, and an F-value of 33.9814. The Significance F value (p-value) is 6.245E-07, indicating that the overall regression model is statistically significant.
  • Residual (df = 15): The error in the model with an SS of 28.4751 and MS of 1.8983.
  • Total (df = 18): Sum of Squares total is 222.

3. Coefficients:

For each predictor:

  • Intercept: Coefficient = 0.2385, not statistically significant (p = 0.7430).
  • Variable 9: Coefficient = 0.2626, statistically significant (p = 0.0325), indicating it contributes to the model.
  • Variable 4: Coefficient = 0.1789, not statistically significant (p = 0.4042).
  • Variable 1156: Coefficient = -0.0014, statistically significant (p = 5.9671E-05), indicating a significant but small effect.

Would you like further details on any part of this analysis?

  1. What does the "Significance F" value imply about the model?
  2. How does Adjusted R Square differ from R Square?
  3. Why is the coefficient for Variable 1156 so low but significant?
  4. How would removing insignificant variables affect the model?
  5. What are possible interpretations of the standard error in this context?

Tip: In regression analysis, a low p-value (< 0.05) suggests that the predictor is statistically significant in explaining the dependent variable.

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Math Problem Analysis

Mathematical Concepts

Regression Analysis
ANOVA
Statistical Significance
Multiple Regression

Formulas

Coefficient calculation
p-value interpretation
R-squared
Adjusted R-squared

Theorems

ANOVA F-test
Statistical Significance Testing

Suitable Grade Level

College/University

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