Abstract
With an increase in the scare of global warming, scientists have proven that temperatures
around the world have been increasing. Not only has the average global temperature increased in the
past 50 years, but the hottest day of the year has shifted nearly two days earlier, according to a new
study by scientists from the University of California, Berkeley, and Harvard University. The
researchers also found that the difference between summer and winter land temperatures has decreased.
With this knowledge, we are interested in whether the temperatures in the winter affect the
temperatures in the summer. We will take a random sample of 40 different years between 1900 and
2008 in Flagstaff, Arizona, Anchorage, Alaska, and Orlando, Florida. We will then measure the
maximum and minimum for each year of our choosing, and also the average winter temperature and the
average summer temperature for each year. From this data, we will be able to predict the mean
temperature of the summer of 2009 and whether it will be warmer than past summers.
In our research we looked at 40 years of data, 1960-2000, in Flagstaff, AZ, Orlando, FL,
Anchorage, AK. Our null hypothesis was that there would be no correlation between summer and
winter temperatures in these three cities. Our alternate hypothesis stated that there is a positive linear
relationship between winter and summer temperatures.
Explanation of Data
Our data is a summary of maximum and minimum temperatures of summer and winter for
Flagstaff, Orlando, and Anchorage. Because we could not take a random sample of years that would be
large enough, we used the 40 years, 1960-2000, that we could find. (
). We analyzed our data by
doing a five number summary of the maximum and minimum temperatures of summer and winter for
all 40 years in all three cities. From there we performed a linear regression t-test to verify the
significance of our hypothesis.
Analysis of Data
Linear Regression t-test
Anchorage, AK Maximum:
Assumptions
1.Linear scatter plot of residuals vs. explanatory (x) should be evenly scattered
*Data is slightly curved but due to the nature of the data (which bunches)
we will proceed with a linear regression t-test .
2. SRS?
*Because of limitations we were unable to obtain an SRS. Proceed with
linear regression t-test.
3. Independence?
*n 10% of weather.
4. Normality?
*Normal Quantile Plot of residuals is fairly linear.
Null Hypothesis: =0
Alternate Hypothesis: 0
Test results: t= 11.72
P= 8.44 E ^-22
R^2 = 0.537
R= 0.733
Df= 118
= 0.05
If in fact there is no correlation between the winter and summer maximum temperatures we are
unlikely to obtain these results. The data is significant at =0.05, we can reject our null hypotheses in
favor of our alternate hypothesis that there is a positive relationship between the winter and summer
maximum temperatures in Anchorage, AK.
Anchorage AK Minimums
*Assumptions and hypothesis stated above.
Test results: t=15.12
P= 1.13E^-29
R^2=0.66
R= 0.81
Df=118
=0.05
If in fact there is no correlation between the winter and summer minimum temperatures we are
extremely unlikely to obtain these results. The data is significant at =0.05. We can reject our null
hypothesis in favor of our alternate hypothesis that there is a positive correlation between winter and
summer minimum temperatures in Anchorage, AK.
Flagstaff, AZ maximum
Assumptions:
1.Linear scatter plot of residuals vs. explanatory (x) should be evenly scattered
*Data is slightly curved but due to the nature of the data (which bunches)
we will proceed with a linear regression t-test .
2. SRS?
*Because of limitations we were unable to obtain an SRS. Proceed with
linear regression t-test.
3. Independence?
*n 10% of weather.
4. Normality?
*Normal Quantile Plot of residuals is fairly linear.
Null hypothesis:=0
Alternate hypothesis:0
Test results: t=9.01
P=6.43E^-13
R^2= 0.58
R=0.76
Df=58
=0.05
If in fact there is no correlation between the winter and summer maximum temperatures we are
extremely unlikely to obtain these results. The data is significant at =0.05. We can reject our null
hypothesis in favor of our alternate hypothesis that there is a positive correlation between winter and
summer maximum temperatures in Flagstaff, AZ.
Flagstaff Minimum
Assumptions and hypotheses stated above
Test results: t=27.44
P=3.22E^-35
R^2=0.93
R=0.96
Df=58
=0.05.
If in fact there is no correlation between the winter and summer minimum temperatures, we are
unlikely to obtain these results. The data is significant at =0.05, and we can reject our null hypothesis
in favor of our alternate hypothesis that there is a positive relationship between the winter and summer
minimum temperatures in Flagstaff, AZ.
Orlando, FL Maximum
Assumptions:
1.Linear scatter plot of residuals vs. explanatory (x) should be evenly scattered
*Data is slightly curved, but due to the nature of the data (which bunches)
we will proceed with a linear regression t-test .
2. SRS?
*Because of limitations we were unable to obtain an SRS. Proceed with
linear regression t-test.
3. Independence?
*n 10% of weather.
4. Normality?
*Normal Quantile Plot of residuals is fairly linear.
Null hypothesis:=0
Test results: t=1.26
P=0.10
R^2= 0.01
R= 0.12
Df=118
= 0.05
Alternate hypothesis:0
If in fact there is no correlation between the winter and summer maximum temperatures, we are likely
to obtain these results. The data is not significant at 0.05. We must fail to reject our null hypothesis
that there is no correlation between winter and summer maximum temperatures in Orlando, FL.
Orlando, FL Minimums
Assumptions and hypotheses stated previously
Test results: t=-0.18
P=0.57
R^2=2.99
R=0.02
Df= 118
=0.05
If in fact there is no correlation between the winter and summer minimum temperatures, we are likely
to obtain these results. The data is not significant at =0.05. We must fail to reject our null hypothesis
that there is no correlation between winter and summer minimum temperatures in Orlando, FL.
Problems
*We could not find data for world cities, so we had to use data from strictly US cities
*We could not find enough data to take a random sample
*We could not find data dating back to the 1900s, so we had to use data ranging from 1960-2000
*We could only find minimum and maximum averages but not the overall means for the day or month
*We could not perform a Chi Squared Test because our data was not in counts
*If we did a follow up project, we would like to have more years and more cities to draw random
samples from.
Because of the fact that we were only able to collect data from three cities and 40 different years, we
cannot state indefinitely that there is an actual correlation between summer and winter temperatures
due to time limits which did not allow us to collect more data. If we had more time, we could have
preformed more tests to support our conclusion.
Conclusion
Based on our linear regression t-tests for each city, we can conclude that winter temperatures in
Flagstaff, AZ and Anchorage, AK have an effect summer temperatures for those cities. It is possible to
say this about the minimum and maximum temperatures for winter and summer in both of these cities
due to the fact that the data is extremely significant at our set alpha level. This significance allows us to
accept our alternate hypothesis that there is a positive linear relationship between summer and winter
temperatures. However, the t-test for Orlando, FL shows that winter temperatures do not have an effect
on summer temperatures. This is because the p-value for summer and winter temperatures is not
significant at our set alpha level, and we fail to reject our null hypothesis that there is no correlation.
Discussion
Further research would require testing different cities, countries, and climates around the world.
Other research could also include comparing precipitation amounts to temperatures, or elevation level
to temperatures.
Appendix
*Data
*Graphs
*Technology used:
*TI-84 Plus calculator
*TI-Connect software
*OpenOffice.Org
*Adobe Photoshop CS3