 Okay, well let's continue with this analysis. Now what I'm going to do here is plot instead the temperature as a function of year. That's plot number one and we no longer want a trend line here. That's blue. That's the state college December temperatures and now for plot number two, I am going to plot the NENU 3.4 index for that year and I'll use axis B to put them on the same scales. So here we can see the two series, blue, the December temperatures, red is the NENU 3.4 index and you can see that in various individual years there does seem to be a relationship where large positive departures of the NENU 3.4 index are associated with warm December's and large negative departures are associated with cold temperatures. So we can see visually that relationship that 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 time series of the two data sets and we can see some of that positive covariance, if you will, that there does appear to be a positive relationship although we already know it's a fairly weak relationship. Now let's do the formal regression. So I'm going to take away the El Nino series. So here we've got state college December temperatures in blue. Now our regression model is going to use the NENU 3.4 index as our independent variable. As a predictor of state college December temperatures, our dependent variable will run the linear regression. Here's the slope 0.74 is the coefficient that describes the relationship in how temperature depends on the NENU 3.4 index. It's positive. We already saw that the slope was positive. There's also a constant term that we're not going to worry about too much here. What we're really interested in is the slope of the regression line that describes how changes in temperature depend on changes in the NENU 3.4 index. As we've seen, that's 0.74 implies that for a unit increase in NENU 3.4 in an anomaly of plus one on the NENU 3.4 scale, we get a temperature for December that on average is 0.74 degrees Fahrenheit warmer than average. The R squared value is 0.0301. Well, if we take 0.0301, 0.0301, and take the square root of that, that's the 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 significance criterion hypothesis test, that is to say, if we didn't know our priority, whether we had reason to believe that El Nino's warm or cool state college December temperatures, then the relationship would not quite be statistically significant. Okay, so we've calculated the linear model. We can now plot it. So now I'm going to plot year and model output on the same scale, and so now the red curve is showing us the component, a 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. The standard deviation is close to...