Question

The data below represents the bacterial growth in a liquid culture over a number of days.

Day	0	4	8	12	16	20
Amount ×106	67	84	98	125	149	185

Find a best-fit equation to the above data trend using polynomial regression with parabolic model given as:

? = ?_? + ?₁? + ?₂?²

After calculating the values of a_o, a₁, and a₂, substitute these values in the above parabolic model. Plot the given data and the obtained parabolic model in the same plot area. Then predict the amount of bacteria after 35 days. Also find standard deviation S_y, standard error of the estimate S_y/x, and correlation coefficient r.

Answer 1

We’ll do all numeric calculation in R. Please ask if you don’t understand any piece of code in the comments below.

Code:

day <- seq(0, 20, by=4)

amount <- c(67,84,98,125,149,185)

fit = lm(amount ~ day + day^2)

summary(fit)

plot(fit)

The result of the regression is:-

Call:
lm(formula = amount ~ day + day^2)

Residuals:
   1    2    3    4    5    6 
 7.0  0.8 -8.4 -4.6 -3.8  9.0 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  60.0000     5.5686   10.78 0.000421 ***
day           5.8000     0.4598   12.61 0.000227 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.694 on 4 degrees of freedom
Multiple R-squared:  0.9755,    Adjusted R-squared:  0.9693 
F-statistic: 159.1 on 1 and 4 DF,  p-value: 0.0002274