Confidence Intervals for Regression In this set of notes we are going to look at confidence intervals that arise from a simple linear regression analysis. Once again, let’s consider the example relating a lion’s proportion of nose blackness to its age. We have already obtained the following parameter estimates which were used to identify the regression equation: Ê (Proportion of nose blackness |Age) = 0.086 + 0.046*Age Confidence intervals for the parameter estimates (slope & intercept) We can use JMP to find the 95% confidence intervals for the regression parameter estimates (intercept and slope). To do this, put your cursor over the output in the Parameter Estimates section and rightclick. Then choose Columns Lower 95% and Upper 95% as shown below. 1 You should get the following output: The details behind these calculations are as follows. Parameter Estimate ± t-quantile*SE(estimate) Degrees of freedom for the t-distribution are df = n – 2 (this is also the df for error in the ANOVA table) To find the t-quantile in JMP for a 95% confidence interval, create a new JMP file and click in the first row to create an observation. Right click on the first column heading and choose Formula. Under Functions (grouped) select Probability t Quantile. Enter the following: Click OK and JMP returns the following. Finally, we can compute the 95% confidence interval for the intercept. 2 We can also compute the 95% confidence interval for the slope. Questions: 1. Interpret the 95% confidence interval for the intercept. 2. Interpret the 95% confidence interval for the slope. 3. Do these confidence intervals agree with the hypothesis test carried out concerning the intercept and slope in the last set of notes? Explain. 3 Confidence interval for the average or mean proportion of nose blackness for a given age How would we estimate the mean proportion of nose blackness of all lions that are 4 years-old? Ê (Proportion of nose blackness |Age) = 0.086 + 0.046*Age Note that if we take another sample of lions, our estimate for this mean will most likely change based on our new regression coefficients. So, to make inferences concerning 4 year-old lions, we should use a ___________________________ for this mean. Computing this interval by hand is beyond the scope of this course, so we will use JMP. To find confidence intervals for the mean proportion of nose blackness associated with ages observed in the data set, first fit the regression model using the Fit Model menu. Then, right-click on the red drop-down arrow next to Response … and select Save Columns Mean Confidence Interval. JMP adds the upper and lower endpoints to the data set. 4 Prediction interval for the proportion of nose blackness for a single lion of a given age How would we estimate the proportion of nose blackness of a single randomly selected lion that is 4 years-old? Ê (Proportion of nose blackness |Age) = 0.086 + 0.046*Age Once again, note that this estimate would likely change based on another sample, so we should obtain a _______________________ to make inferences about the proportion of nose blackness for a single 4 year-old lion. You can use JMP to calculate this prediction interval. After fitting the regression model, right-click on the red drop-down arrow next to Response … and select Save Columns Indiv Confidence Interval. JMP adds the upper and lower endpoints to the data set. 5 Questions: 1. How did your point estimate of the mean proportion of nose blackness of all 4-year-old lions compare to your point estimate of the mean proportion of nose blackness of a single randomly selected 4-year-old lion? 2. Interpret the 95% confidence interval for the mean proportion of nose blackness for all 4-yearold lions. 3. Interpret the 95% prediction interval for the proportion of nose blackness for a single 4-year-old lion. 4. How do these two intervals relate to each other, and why does this make sense? 6

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