To determine the 95% confidence interval for the slope in a linear regression analysis, you'll need the following information:
- Slope (\(b_1\)) of the regression line.
- Standard error (SE) of the slope.
- The number of data points (\(n\)).
Given that \(n = 20\), the degrees of freedom (df) will be \(n - 2 = 18\).
The formula for the 95% confidence interval for the slope is:
\[ b_1 \pm t_{\frac{\alpha}{2}} \times SE(b_1) \]
Where:
- \(b_1\) is the estimated slope.
- \(t_{\frac{\alpha}{2}}\) is the critical t-value for the desired confidence level and degrees of freedom.
- \(SE(b_1)\) is the standard error of the slope.
Steps to Calculate the Confidence Interval:
- Find the estimated slope (\(b_1\)) and standard error of the slope (\(SE(b_1)\)) from your regression analysis.
- Find the t-value from a t-distribution table for \(df = 18\) and \( \frac{\alpha}{2} = 0.025\) (for a two-tailed test). The critical t-value (for df = 18) is approximately 2.101.
- Calculate the confidence interval using the formula above.
Example Calculation:
- Suppose \(b_1 = 3.5\) and \(SE(b_1) = 0.75\).
Calculating the confidence interval:
\[ \text{Lower Limit} = 3.5 - (2.101 \times 0.75) \] \[ \text{Upper Limit} = 3.5 + (2.101 \times 0.75) \]
Calculate the product:
\[ 2.101 \times 0.75 \approx 1.576 \]
Now substituting back:
\[ \text{Lower Limit} \approx 3.5 - 1.576 \approx 1.924 \] \[ \text{Upper Limit} \approx 3.5 + 1.576 \approx 5.076 \]
Therefore, the 95% confidence interval for the slope would be approximately \((1.924, 5.076)\).
Conclusion:
To compute your specific confidence interval, you will need to substitute the actual values of your slope estimate and its standard error into the formula provided. If you can provide those values, I can help you calculate the exact confidence interval.
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