A researcher felt there might be a correlation between weather conditions and people's mental health. In order to verify his conjecture, he obtained data on the number of mostly sunny days (x) in 2015 from 245 randomly selected cities in the nation and their corresponding citizens' happiness indices (y). In other words, he wants to test if there is a statistically significant correlation between x and y. How should he carry out the analysis?
1) Run a differences-in-means test (also sometimes called here a "paired t-test") on x and y with the null hypothesis that their means are equal and see if the resulting p-value is small for appropriately phrased hypotheses
2) Run a simple linear regression and test if the slope estimate is statistically different from 1 with a t-test
3) Run a simple linear regression and test if the slope estimate is statistically different from 0 with a t-test
4) Run a simple linear regression and check if r2 is bigger than 0.5
5) Run a t-test on y with the null hypothesis that the true mean of y equals the sample mean of x