Exponential Regression model

Question

I have no idea how to complete the below problem and teacher has not gotten back to me. Can someone please show me how to complete this?

 

The data below shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup. The brewing pot temperature is approximately 180 degrees (F).

 

Time (mins)     Temp (Degree F)

       0                     179.5

       5                     168.7

       8                     158.1

       11                   149.2

       15                   141.7

       18                   134.6

       22                   125.4

       25                   123.5

       30                   116.3

       34                   113.2

       38                   109.1

       42                   105.7

       45                   102.2

       50                   100.5

 

a. Find an exponential regression model of the form y=a*b^x to represent the above date, where x is the number of minutes, and y is the temperature of the cup of coffee.

 

b. Graph the exponential regression model.

 

c. Decide whether the model is a "good fit" to represent this data.

 

d. When is the coffee at a temperature of 92 degrees?

 

e. What is the predicted temperature of the cup of coffee after 65 minutes? 

Answer 1

Hi Jeff,

 

To find the best fit exponential regression model describing this set of data - you can use the statistics function on either a scientific or a graphing calculator.

 

When inputting the time (minutes) data values into list 1 and their corresponding temperature (degrees) data values into list 2, these were the values given by the calculator for the exponential regression coefficients:

    a = 171.46

    b = 0.988

 

Below is the link to our graph of our exponential regression model:

 

https://dl.dropbox.com/s/r7ofylsarzx05q1/Gragh_Exponential_Regression_Model.png?raw=1

 

The graph was generated on the Casio Prizm graphing calculator, but could also have been easily produced on the TI 8x series of graphing calculators used by most schools.

 

The regression coefficient (r) is used to determine whether or not our set of data is a good exponential curve fit.  The calculator returned an (r) value of -0.985.  The range of possible (r) values is -1 to 1 having these value interpretations:

    -1 (high negative correlation, i.e. very good fit)

     0 (no correlation, i.e. very bad fit)

     1 (high positive correlation, i.e. very good fit)

 

Our value of -0.985 shows a very good fit to describe our decreasing temperature values vs. our increasing time values.  This is confirmed by the smooth slight downward curve from left to right shown on our graph.

 

We can calculate the time it takes the coffee to reach a temperature of 92 degrees using our exponential regression equation:

    y = 171.46(0.988)^x

    92 = 171.46(0.988)^x

    0.988^x = 92/171.46

    0.988^x = 0.537

    xlog(0.988) = log(0.537)

    x = log(0.537)/log(0.988)

    x = 52 degrees

This value is confirmed on our graph.

 

We can also calculate what would be the predicted temperature after the coffee was allowed to brew for 65 minutes using our exponential regression equation:

    y = 171.46(0.988)^x

    y = 171.46(0.988)⁶⁵

    y = 171.46(0.456)

    y = 78 degrees

Once again this value is confirmed on our graph (slight difference is a rounding issue).

 

Our graph also shows the y-intercept (the regression coefficient (a) value).  This is the maximum temperature value of the coffee before it was allowed to brew.

 

Trust the above helps your understanding concerning the graphical interpretation of our exponential regression model.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.