 Now we're going to look at a somewhat different situation where our independent variable is no longer time, but it's some quantity It could be temperature. It could be an index of El Nino or the North Atlantic Oscillation Let's look at an example of that sort We are going to look at the relationship between El Nino and December temperatures in State College, Pennsylvania And we can plot out that relationship as a scatter plot On the y-axis we have December temperature in State College The x-axis is our independent variable the Nino 3.4 index Negative values indicate La Nina and positive values indicate El Nino's And the strength of the relationship between the two is going to be determined by the trend line That describes how December temperatures in State College depend on El Nino and by fitting that regression We obtain a slope of 0.7397 That means for each unit change in El Nino in Nino 3.4, we get a 0.74 unit change in temperature So for a moderate El Nino event where the Nino 3.4 index is in the range of plus one That would imply that December temperatures in State College for that year are 0.74 degrees Fahrenheit 0.74 degrees Fahrenheit warmer than usual And for a modestly strong La Nina where the Nino 3.4 index is on the order of minus one or so The December State College December temperatures would be about 0.74 degrees colder than normal You can also see that the y-intercept here the case where the Nino 3.4 index is zero We get roughly the climatological value for December temperatures 30.9 Now the correlation coefficient is associated with that linear regression in this case is 0.174 Now we have 107 years our data set as before goes from 1888 to 1994 if we use our table and take n equal to 107 and r of 0.174 We find that the one tailed value of p is 0.0365 the two tailed value is 0.073 So if our threshold for significance were p of 0.05 the 95 percent significance level Then that relationship a correlation of coefficient of 0.174 with a hundred and seven years of information Would be significant for one tailed test But it would not pass the 0.05 the 95 percent significance threshold for two tailed test So we have to ask the question which is more appropriate here the one tailed test or the two tailed test Now if you had a reason to believe that El Nino events warm the northeastern US for example You might motivate a one-tailed test since only a positive relationship would be consistent with your Expectations, but if we didn't know beforehand whether El Nino's had a cooling influence or warming influence on the northeastern US You might argue for a two-tailed test So whether or not the relationship is significant at the p equals 0.05 level is going to depend on which type of hypothesis test we're able to use in this case