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QUESTION: An experiment was conducted to ascertain which of two bacteria, X or Y, would grow best using lactose as a substrate. Four cultures each of Bacterium X and Bacterium Y were grown and produced the biomass results (in milligrams) shown in the table below. Which bacterium, X or Y, had a higher mean biomass when grown using lactose as the substrate, and was the difference significant?

Bacterium X 
Bacterium Y 
Flask A 
501 
233 
Flask B 
468 
270 
Flask C 
523 
280 
Flask D 
460 
249 
ANSWER:
In order to compare the means for X and Y, we will use Student’s ttest. Essentially, the ttest compares the difference between the two means to the overall variation in the data—the standard deviation of the difference between the means. To compute the necessary t statistic, we must first find the means for X and Y, the variance for X and for Y (and by extension overall variance), and the mean difference between X and Y. We then look up this t value in a table for the tdistribution with 6 degrees of freedom (nx + ny – 2) and twotails, giving the probability P that the mean difference between X and Y is due to chance. The calculations are shown below.
Bacterium X 
Bacterium Y 

Σx 
1952 
1032 
First we calculate the sum of the values 
n 
4 
4 
n is simply the number of samples for 
x̄ 
488 
258 
Next, we calculate the mean biomass 
Σx2 
955154 
267590 
Σx2 signifies the sum of the values 
(Σx)2 
3810304 
1065024 
(Σx)2 signifies the totals for X and Y 
(Σx)2/n 
952576 
266256 

Σd2 
2578 
1334 
Σd2= Σx2(Σx)2/n , where Σd2 is the 
σ2 
859.3 
444.7 
σ2 = Σd2/(n1) , where σ2 is the variance 
σd2 
326 
σd2 = σx2/nx + σy2/ny , where σd2 is 

σd 
18.1 

t 
12.7 
t = (x̄x  x̄y)/σd 

P 
0.00001436 
From the calculations above, the probability that we would observe the mean difference found between X and Y purely by chance is 0.001436%. That value is lower than the conventional 5% and 1% thresholds for significant and highly significant data respectively. Thus, we can confidently claim that Bacterium X grew better than Bacterium Y using lactose as a substrate!