Statistics - Ti 83 Exponential Regression



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Ti 83 Exponential Regression is used to compute an equation which best fits the co-relation between sets of indisciriminate variables.

Formula

${ y = a \times b^x}$

Where −

  • ${a, b}$ = coefficients for the exponential.

Example

Problem Statement:

Calculate Exponential Regression Equation(y) for the following data points.

Time (min), Ti051015
Temperature (°F), Te140129119112

Solution:

Let consider a and b as coefficients for the exponential Regression.

Step 1

${ b = e^{ \frac{n \times \sum Ti log(Te) - \sum (Ti) \times \sum log(Te) } {n \times \sum (Ti)^2 - \times (Ti) \times \sum (Ti) }} } $

Where −

  • ${n}$ = total number of items.

${ \sum Ti log(Te) = 0 \times log(140) + 5 \times log(129) + 10 \times log(119) + 15 \times log(112) = 62.0466 \\[7pt] \sum log(L2) = log(140) + log(129) + log(119) + log(112) = 8.3814 \\[7pt] \sum Ti = (0 + 5 + 10 + 15) = 30 \\[7pt] \sum Ti^2 = (0^2 + 5^2 + 10^2 + 15^2) = 350 \\[7pt] \implies b = e^{\frac {4 \times 62.0466 - 30 \times 8.3814} {4 \times 350 - 30 \times 30}} \\[7pt] = e^{-0.0065112} \\[7pt] = 0.9935 } $

Step 2

${ a = e^{ \frac{\sum log(Te) - \sum (Ti) \times log(b)}{n} } \\[7pt] = e^{\frac{8.3814 - 30 \times log(0.9935)}{4}} \\[7pt] = e^2.116590964 \\[7pt] = 8.3028 } $

Step 3

Putting the value of a and b in Exponential Regression Equation(y), we get.

${ y = a \times b^x \\[7pt] = 8.3028 \times 0.9935^x } $



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