Time and Temperature

Part 3: Fitting to Temperature Data

The following table shows long-term average temperatures in Kalamazoo, Michigan. (Source: Core-Plus Mathematics Project.) In this Part we determine whether such data can be fit with a sinusoidal function.

1. Enter the data in your worksheet, and make a scatter plot.
2. Estimate the coefficients A, B, C, and t₀ of a function of the form
   T(t) = A + B sin [ C (t - t₀)]

   that should fit this data. Plot your sinusoidal function on the data. How good is the fit? Adjust the coefficients as necessary to make the fit as good as you can.
3. Finland has a delightful web site to inform travelers about the Finnish weather. (Sorry, link is broken.) It includes normal ranges of temperatures for Helsinki for each month of the year. If we average the normal highs and lows for each month, we can derive the following table of data. Decide whether these data are approximately sinusoidal, and find the best fitting sinusoid you can.

4. Every Chamber of Commerce in the country seems to have a web site that includes data on coldest and warmest months. The following table is a sampling of such data. Choose one location, and construct a sinusoidal function that should approximate long-term average temperature at that location. Then use your function to estimate average temperatures for one Spring month and one Fall month.