## A broth used to manufacture a pharmaceutical product has its sugar content, in mg/mL, measured several times on each of three successive day

Question

A broth used to manufacture a pharmaceutical product has its sugar content, in mg/mL, measured several times on each of three successive days. Day 1: 5.0 4.8 5.1 5.1 4.8 5.1 4.8 4.8 5.0 5.2 4.9 4.9 5.0 Day 2: 5.8 4.7 4.7 4.9 5.1 4.9 5.4 5.3 5.3 4.8 5.7 5.1 5.7 Day 3: 6.3 4.7 5.1 5.9 5.1 5.9 4.7 6.0 5.3 4.9 5.7 5.3 5.6 Can you conclude that the variability of the process is greater on the second day than on the first day?

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2 weeks 2021-11-22T04:23:06+00:00 1 Answer 0

1. Answer: Yes, the second day has more variability than the first day.

Step-by-step explanation:

Day 1: 5.0 4.8 5.1 5.1 4.8 5.1 4.8 4.8 5.0 5.2 4.9 4.9 5.0

Day 2: 5.8 4.7 4.7 4.9 5.1 4.9 5.4 5.3 5.3 4.8 5.7 5.1 5.7

Day 3: 6.3 4.7 5.1 5.9 5.1 5.9 4.7 6.0 5.3 4.9 5.7 5.3 5.6

The variability is related to the standard deviation, so we need to compare the standar deviation of the second day with the standard deviation of the first day.

The standard deviation can be calculated with a scientific calculator, and the method will depend on the one you are using.

Now, we also could do it by hand:

S = √( (1/(N-1))*∑(xₙ – xm)^2)

where N is the number of points (13 in both days), xm is the mean of the set and xₙ is each point of the set:

the means for both sets are:

xm1 = 4.9615

xm2 = 5.1846

S1 = 0.1387

S2 = 0.3848

Here we can see that the standard deviation in the second day is bigger, this would mean that the second distribution is less “thight” than the first one, so the second day has more variability than the first day.