 We're going to look at an example here. I'm loading in December average temperatures for state college Pennsylvania for the 107 year period from 1888 to 1994. Let's plot out the data. That's what they look like. It's a scatter plot. Or if we like, we can view them in terms of a line plot by clicking this radio button. If we look at the statistics tab, it tells you the average of the temperature is 30.9. Just under 31 degrees Fahrenheit. So the average December temperature in state college of Pennsylvania is about a degree Fahrenheit below freezing. The standard deviation is 3.95, just under 4 degrees Fahrenheit. So the fluctuation from year to year in the average December temperature in state college is a fairly sizable 4 degrees Fahrenheit. One year it might be 30. The next year it might be 34. The next year it might be 31. The next year it might be 27. That gives you some idea of the fluctuations. And of course we can see those fluctuations here in the plot. Now we can calculate a trend line. Let's go to the trend lines tab. This calculates a linear trend in the time series. It tells us there's a trend right here of 0.025 degrees Fahrenheit warming per year. Or if we want to express that in terms of a century, 2.5 degrees Fahrenheit warming per century. That's the warming trend in state college Pennsylvania. Now the correlation for that coefficient for that regression R is R. It equals 0.193. We look that up in the online statistics table. I put in 107 years for the length of our series and 0.193 for R, it calculate. We look that up in the online statistics table. We calculate significance. It tells us the P value is 0.023 for one-tailed tests and 0.046 for two-tailed tests. So in either case, the correlation, the regression, the trend in the series will be significant at greater than P equals 0.05 level. It would be significant at the 95% confidence level. Arguably we should go with the one-tailed test since we're really testing the hypothesis that there is a warming trend in state college. Since we know the globe is warming, our hypothesis was unlikely to be that state college showed a cooling trend. We are interested to see if state college showed the warming trend that we know is evident in the temperature records around the world. So one could, in fact, one typically would motivate a one-tailed or one-sided hypothesis test. So the trend passes that test at the 0.02 level. That's fairly significant. If we go back again, we can see the standard error of the slope is 0.012. If we were to take the value 0.025 and add plus or minus two times this number of 0.012, it would give us the 95% confidence range in the slope of this warming trend.