Introduction

Observational evidence indicates that Wisconsin winters are exhibiting a warming trend, suggesting a temporal correlation with temperature. Through a linear regression analysis of data from the NOAA Climate Divisional Database, a positive linear relationship between year and temperature in Wisconsin was identified.

Background

The National Oceanic and Atmospheric Administration’s Climate Divisional Database (nClimDiv) collected monthly temperature data between 1895 and 2018 for states in the U.S., not including Hawaii, and data starting in 1925 for Alaska. This dataset (climdiv_state_year.csv) is based on the Global Historical Climatological Network-Daily (GHCN-D). The data were collected using an enhanced methodology involving multiple processes and tools. The primary input came from the Global Historical Climatological Network-Daily (GHCN-D), supplemented by additional station networks to improve spatial coverage and data reliability. The raw data contained data from all of the states in the USA (not including Hawaii), as well as the temperature in both Fahrenheit and Celsius. For this project, I selected just the data from Wisconsin (fips=55), and only included the temperature in Fahrenheit.

Graph

The graph below shows the Average Annual Temperature per year in Wisconsin. It is clear to see that the correlation line trends up and to the left, suggesting a positive correlation between temperature and year.

The graph does demonstrate a positive correlation; however, more statistical analysis is needed to confirm our prediction.

Statistical Analysis

Step 1: Model Statement

Parameter of Interest: X = temperature (in Fahrenheit) Type of Inference: Linear Regression

\(X_i = \beta_0 + \beta_1 * X + \epsilon_i\), for \(i = 1,...n\) where \(\epsilon_i \sim N(0, \sigma)\)

Linearity is satisfied. The residual plot does not show any obvious curve pattern.

Normal errors around 0 is satisfied, the residuals tend towards the central line. Large outliers are uncommon, and they are relatively symmetric.

Constant variance is satisfied. The residuals do not obviously funnel in as you move horizontally on the graph.

Step 2: State Hypotheses

\[H_0: \beta_1 = 0\] \[H_{alpha}: \beta_1 > 0\] To investigate the presence of a positive linear relationship between year and temperature, a one-sided alternative hypothesis is employed. The null hypothesis assumes no relationship, thereby testing whether temperature increases as the year progresses.

Step 3: Identify Test Statistic and Null Sampling Distribution

The test statistic (and null distribution) for a hypothesis test for \(\beta_1\) is: \[T = \frac{\hat{\beta_1} - \beta_{1, null}}{SE(\hat{\beta_1})} \sim t(n - 2)\]

Step 4: Identify Outcomes From Data and Alternative Hypothesis

## [1] 4.347995

The test statistic is 4.347995.

Step 5: Calculate P-Value

## [1] 1.409927e-05

The one-sided p-value is 1.409927e-05.

Step 6: Interpret in Context

For \(\alpha\) = 0.05, reject the null hypothesis because 1.409927e-05 < 0.05. There is strong evidence for a positive linear relationship between year and temperature.

Report

Further Interpretation

The p-value got from analysis is 1.409927e-05. By comparing this value to an alpha of 0.05, 1.409927e-05 is less than 0.05. So, we can rejecet the null hypothesis. This means that there is strong evidence for a positive linear relationship between year and temperature in Wisconsin.

Conclusion

Through linear regression and a one-tailed test, statistically significant evidence was obtained to support the existence of a positive relationship between year and temperature in Wisconsin.