Math Problem Statement
Description: The following is an accessible version of the Python script output for OLS Regression. In the normal output, the items are arranged in two columns. Here, items are listed in one column for easier use. Each item has a label. For example, the label “Model” comes before the item “OLS”. Items follow the same order as the normal Python script output. OLS Regression Results Dependent Variable: Quality
Model: OLS
Method: Least Squares
Date: Sun, 18 Aug 2019
Time: 11:39:31
No. Observations: 18
Df Residuals: 15
Df Model: 2
Covariance Type: nonrubust
R-squared: 0.978
Adj. R-squared: 0.975
F-statistic: 332.2
Prob (F-statistc): 3.80 e -13
Log-Likelihood: -21.142
AIC: 48.28
BIC: 50.95 **** Coef
Std err
T
P>|t|
[0.025
0.975]
Intercept
0.5382
0.473
1.137
0.273
-0.471
1.547
Speed
-1.9046
0.176
-10.834
0.000
-2.279
-1.530
Angle
4.0280
0.178
22.574
0.000
3.648
4.408 Omnibus: 4.358
Prob(Omnibus): 0.113
Skew: 0.082
Kurtosis: 1.637
Durbin-Watson: 2.121
Jarque-Bera (JB): 1.414
Prob(JB): 0.493
Cond. No: 14.4
The statsmodels ols() method is used on a cars dataset to fit a multiple regression model using Quality as the response variable. Speed and Angle are used as predictor variables. The general form of this model is:
If the level of significance, alpha, is 0.05, based on the output shown, what is the correct interpretation of the overall F-test? (Hint: Prob (F-statistic) is the P-value). Select one.
the answers: Null Hypothesis Alternative Hypothesis P-value = 332.2 Since the P-value is greater than the level of significance, alpha, do not reject the null hypothesis. Conclude that all beta values are zero. Therefore, the model is not statistically significant.
Null Hypothesis Alternative Hypothesis P-value = 3.80e-13 Since the P-value is less than the level of significance, alpha, reject the null hypothesis. Conclude that at least one beta value is non-zero. Therefore, the model is statistically significant.
Null Hypothesis Alternative Hypothesis P-value = 3.80e-13 Since the P-value is less than the level of significance, alpha, do not reject the null hypothesis. Conclude that all beta values are zero. Therefore, the model is not statistically significant.
Null Hypothesis Alternative Hypothesis P-value = 332.2 Since the P-value is greater than the level of significance, alpha, reject the null hypothesis. Conclude that at least one beta value is non-zero. Therefore, the model is statistically significant.
Solution
The correct interpretation of the overall F-test, based on the output provided, is:
Null Hypothesis: All beta values are zero (i.e., none of the predictor variables have any effect on the response variable).
Alternative Hypothesis: At least one beta value is non-zero (i.e., at least one predictor variable has an effect on the response variable).
Since the P-value = 3.80e-13 (which is much smaller than the significance level α = 0.05), we reject the null hypothesis. This means that we conclude that at least one beta value is non-zero, and the model is statistically significant.
Thus, the correct answer is:
Null Hypothesis
Alternative Hypothesis
P-value = 3.80e-13
Since the P-value is less than the level of significance, alpha, reject the null hypothesis. Conclude that at least one beta value is non-zero. Therefore, the model is statistically significant.
Would you like more details on any part of this analysis or further clarification? Here are 5 follow-up questions that might be useful:
- How do I interpret the individual P-values of the predictors (Speed and Angle)?
- What does the R-squared value indicate about the model's fit?
- How can I perform a hypothesis test on the individual coefficients?
- What does the "Durbin-Watson" statistic tell me about the model?
- How can I improve the model if the assumptions are violated?
Tip: The F-test assesses whether the overall regression model is useful, whereas individual hypothesis tests (like t-tests) help evaluate the significance of each predictor in the model.
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Math Problem Analysis
Mathematical Concepts
Statistical Inference
Hypothesis Testing
Linear Regression
Formulas
F-statistic formula: F = (SSR / k) / (SSE / (n - k - 1))
P-value comparison to significance level alpha
Theorems
Null Hypothesis Testing
P-value Interpretation
F-distribution Properties
Suitable Grade Level
Undergraduate Level (Statistics or Data Science)