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Question

Natalie performs a chemistry experiment where she records the temperature of an ongoing reaction. The solution is 93.5°C after three minutes 90°C after five minutes 84.8°C after nine minutes 70.2°C after 18 minutes 54.4°C after 30 minutes 42.5°C after 37 minutes and 24.9°C after 40 minutes. Perform a linear regression on this data to complete the following items.
on average how much does the temperature decrease every five minutes?

Answer 1

The given data shows a range from 93.5^oC to 24.9^oC, which has a Δ of 68.6^oC. That difference occurred over a period of 40 minutes, which yields an average of -1.7^oC per minute.

 

1. A very good generalization can be made as follows: 

When plotted on a graph, the data yield a straight line from 93.5^oC to 24.9^oC with a negative slope. Thus, we can determine the formula for this straight line to be Y = -mX +93.5.

 

When we compose of table of X values from 0 to 40, the results match well with the observations, and yields an average decline of 8.6^oC per 5 minutes..

 

2. To use the linear regression method (least squares method):

we consider the first measurement as time 1, extend the line to the y axis, where we find the intercept is 97.1. The mean value is 61.7.

 

With this information, we can compute the data points by x = X - Xa, where Xa is the mean value for X, and Y = Y -Ya, where Ya is the mean value for Y, and then compute values using the formula: y = (∑xy/∑x²)x

 

Regression line equation: Y= -1.66X + 97.1, and yields an average decline of 8.3^oC per 5 minutes.

By the way, the most influential measurement was #3 (70.2^oC at 18 minutes). In practice, a second set of measurements would be in order.