Question

The population of a culture of bacteria is calculated and recorded after each hour, for a six-hour period.

0	98
1	206
2	405
3	783
4	1550
5	2520
6	5630

1. Sketch a graph to illustrate these data.
2. Compare linear, quadratic, and exponential regression models. List the equations and correlation coefficients for each type of function. Provide a graph and include the regression lines.
3. Use each model to determine the initial number of bacteria at the beginning of the study.
4. Which function is the best model for these data? Explain your reasoning.

Answer 1

The R code given below.

X <- 0:6
Y <- c(98,   206,   405,   783,   1550,   2520,   5630)

linear.model <- lm(Y ~ X)
print(summary(linear.model))
X2 <- X^2
quadratic.model <- lm(Y ~ X+X2)
print(summary(quadratic.model))
exponential.model <- nls(Y ~ exp(a + b * X),start = list(a = 1, b = 1))
print(summary(exponential.model))

plot(1:1)
dev.new()
plot(X,Y, col="blue", lwd=2, xlab="X", ylab = "Y", main="Scatterplot & Regression lines")
curve( 798.9*x-797.8, xlim=c(0,6), lwd=2, col = "red", add=TRUE)
curve(233.8 *x^2-604.2*x+371.4, lwd=2, col = "violet", add=TRUE)
curve(exp(4.4729+ 0.6916*x), lwd=2, col = "green", add=TRUE)

a,b) The scatter plot and regression lines are plotted on the same window.

The linear model is : and Multiple R-squared: 0.7642

The quadratic model is :

and Multiple R-squared: 0.9606

The exponential model is :

The exponential model is the best fit and the linear model is the least fit.

c) The initial number of bacteria at the beginning of the study are calculated below.
For the linear model, For The quadratic model is : :

d) From the plots, The exponential model is the best fit.