 Let's continue with this analysis. Now, what I'm going to do here is plot instead the temperature as a function of year instead of NENIA 3.4. That's plot number one. That's the state college December temperatures. And now for plot number two, I'm going to plot the NENIA 3.4 index as a function of year. And I use axis B here to put them on the same scale. So here we can see the two series. We have the state college December temperatures in blue and the NENIA 3.4 index in yellow. And you can see that in various years, there does seem to be a little bit of a relationship between large positive departures of the NENIA 3.4 index associated with warm December temperatures and large negative departures associated with cold temperatures. We can visually see that relationship. We also saw when we plotted the two variables in a two-dimensional scatter plot and looked at the slope of the line relating the two data sets. Here now, we're looking at the same time, looking at the time series of the two data sets. And we can see some of that positive covariance, if you will, that there does seem to be a positive relationship, although we already know it's a fairly weak relationship. So let's do a formal regression. So I'm going to take away the NENIA series here. What we've got here is our state college December temperatures in blue. Now our regression model is going to use the NENIA 3.4 index as the independent parameter and the temperature as our dependent variable. We'll run the linear regression. Whereas the slope 0.74 is the coefficient that describes the relationship between the NENIA 3.4 index and December temperatures. It's positive. We already saw the slope was positive. There's also a constant term we're not going to worry much about here. What we're really interested in is the slope of the regression line that describes changes in temperature depends on changes in the NENIA 3.4 index. And as we've seen, 0.74 implies that for a unit increase in NENIA 3.4, an anomaly of plus one on the NENIA 3.4 scale will get a temperature for December that on average is 0.74 degrees Fahrenheit warmer than average. The R-squared value right here is 0.0302. And if we take that number, take the square root of that, that's an R value of 0.1734. And we know it's a positive correlation because the slope is positive. We already looked up the statistical significance of that number, and we found that for a one-sided hypothesis test that the relationship is significant at the 0.05 level. But if we were using a two-sided significant criterion hypothesis test, that is to say, if we didn't know a priori, whether we had a reason to believe that NENIA's warm or cool state college December temperatures, then the relationship would not quite be statistically significant. So we've calculated the linear model. So now we can plot it. So now I'm going to plot year and model output on the same scale. You can change the scale of these axes by clicking on these arrows. I'm going to make them both go from 20 to 40. This one over here. And so now the yellow curve is showing us the component of variation in the blue curve that can be explained by El Nino. And we can see it's a fairly small component. It's small compared to the overall level of variability in December state college temperatures, which vary by as much as plus or minus four degrees or so Fahrenheit.