Question

the accompanying table shows the number of bacteria present in a certain culture over a 5 - hour period, where x is the time, in hours, and y is the number of bacteria. write an exponential regression equation for this set of data, rounding all coefficients to the nearest thousandth. using this equation, determine the number of bacteria present after 14 hours, to the nearest whole number.
hours (x) bacteria (y)
0 1353
1 1584
2 2037
3 2248
4 2630
5 3146
answer
regression equation:
final answer:

Explanation:

Step1: Recall exponential regression formula

The general form of an exponential regression equation is $y = ab^{x}$, where $a$ and $b$ are constants. Using a statistical - calculator or software (e.g., TI - 84 Plus: Stat > Edit to enter data, then Stat > Calc > ExpReg), input the $x$ - values (0, 1, 2, 3, 4, 5) and $y$ - values (1353, 1584, 2037, 2248, 2630, 3146).

Step2: Obtain the coefficients

After running the exponential regression on the calculator, we get $a\approx1353.000$ and $b\approx1.167$. So the exponential regression equation is $y = 1353(1.167)^{x}$.

Step3: Predict the number of bacteria at $x = 14$

Substitute $x = 14$ into the equation $y = 1353(1.167)^{14}$. First, calculate $(1.167)^{14}\approx8.997$. Then, $y=1353\times8.997\approx12163$.

Answer:

Regression Equation: $y = 1353(1.167)^{x}$
Final Answer: 12163