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**Statistical Thinking for Industrial Problem Solving**

In this video, we again use the Cleaning data to fit a regression model for **Removal** and **ID** using Fit Y by X. We'll discuss the statistical output provided, and will see how to make predictions using our regression equation.

We'll again use** Removal** as the Y, Response and **ID** as the X, Factor.

We'll select Fit Line from the red triangle to fit the regression model.

In a previous video we conducted a residual analysis and verified that our model assumptions were not violated.

Now, let's take a look at the default statistical output.

The summary statistics for the model fit, including root mean square error and RSquare, are reported in the Summary of Fit table. Our root mean square error is 1.7, and RSquare is 0.64.

Thus, **ID** explains 64% of the variation in **Removal**.

The overall test results for the significance of the model are reported in the ANOVA table. The p-value is very small, so we can conclude that the model is significant.

As we saw earlier, the fitted linear model is reported under linear fit.

The estimated intercept and slope coefficients for this model are reported in the Parameter Estimates table under Estimate.

We might be interested in using the model to predict the average **Removal** for different values of **ID**. As we have seen, we can simply plug a value of **ID** into the equation to calculate the predicted **Removal**.

There are also a number of ways to make predictions directly in JMP.

We can select Save Predicteds from the red triangle for Linear Fit to save the formula for the model to the data table.

This computes the predicted Removal for all of our observations. We can add a new row, and enter a value for **ID**, and JMP will predict the average removal for this value.

Keep in mind that it only makes sense to enter values of the predictor within the range used to build the model. The model may not predict well if we extrapolate beyond the range of our predictor values.

Let's return to the analysis window.

If we're only interested in approximating the mean **Removal**, we can use the cross-hair tool from our toolbar. When we click on the line, the cross-hair tool will display the predicted Removal for a given value of **ID**.

We can also display two types of intervals on our Bivariate Fit Plot: Confidence Intervals and Prediction Intervals.

The Confidence Curves Fit option from Linear Fit displays confidence bands. These bands represent confidence intervals for the mean **Removal** for a given value of **ID**.

For example, the confidence interval for the mean **Removal** for parts with **ID** of 10 is approximately 10.5 to 11.5 units.

The Confidence Curves Indiv (Individual) option displays prediction bands. These bands represent prediction intervals for individual values of **Removal** for given values of **ID**.

For example, the predicted range of values for parts with an **ID** of 10 is approximately 7.5 to 14.5.