a. Is the model as a whole significant? Why or why not? b. Compute the correlation coefficient between poverty and infant mortality. c. Test whether the correlation is significant... a. Is the model as a whole significant? Why or why not? b. Compute the correlation coefficient between poverty and infant mortality. c. Test whether the correlation is significant or not.
Understand the Problem
The question is asking to analyze the output of a regression analysis related to the relationship between poverty and infant mortality. It requires evaluating the significance of the entire model, calculating the correlation coefficient, and testing the significance of that correlation.
Answer
a. Yes, model is significant. b. $R = 0.318580737$. c. Yes, correlation is significant.
Answer for screen readers
a. Yes, the model is significant since the p-value ( (1.96073 \times 10^{-5}) < 0.05 ).
b. The correlation coefficient between poverty and infant mortality is ( R = 0.318580737 ).
c. Yes, the correlation is significant because the p-value for the correlation ( (1.96073 \times 10^{-5}) < 0.05 ).
Steps to Solve
- Evaluate Model Significance The model significance can be evaluated using the F-statistic and its corresponding p-value. The F-statistic is provided as ( F = 22.441 ) and the significance (p-value) is given as ( \text{Significance F} = 1.96073 \times 10^{-5} ).
To determine if the model is significant, we typically use a threshold (commonly ( \alpha = 0.05 )):
- If ( p < 0.05 ), reject the null hypothesis and conclude the model is significant.
- If ( p \geq 0.05 ), do not reject the null hypothesis.
Here, since ( 1.96073 \times 10^{-5} < 0.05 ), the model is significant.
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Compute the Correlation Coefficient The correlation coefficient ( R ) is directly provided as ( R = 0.318580737 ). This value can be interpreted as the strength and direction of the linear relationship between poverty and infant mortality.
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Test the Correlation Significance To test whether the correlation is significant, we can use the t-statistic of the regression coefficient for the variable of interest (in this case, "Infant Mortality"). The t-statistic is given as ( t = 4.737215831 ) with a corresponding p-value ( = 1.96073 \times 10^{-5} ).
Again, we compare the p-value to the significance level:
- If ( p < 0.05 ), the correlation is significant.
- If ( p \geq 0.05 ), the correlation is not significant.
In this case, ( 1.96073 \times 10^{-5} < 0.05 ), indicating the correlation is significant.
a. Yes, the model is significant since the p-value ( (1.96073 \times 10^{-5}) < 0.05 ).
b. The correlation coefficient between poverty and infant mortality is ( R = 0.318580737 ).
c. Yes, the correlation is significant because the p-value for the correlation ( (1.96073 \times 10^{-5}) < 0.05 ).
More Information
The small p-values indicate strong evidence against the null hypothesis, which asserts that there is no relationship between the variables. The correlation coefficient indicates a weak positive linear relationship.
Tips
- Failing to compare p-values correctly to the significance level; ensure the correct interpretation of whether to reject or not reject the null hypothesis.
- Misunderstanding the meaning of the correlation coefficient (confusing it with causation).
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