BIOL 1406
PreLab 1.11
What is the Student’s ttest?
In the previous section, when comparing measurements made using a beaker and graduated cylinder, we asked, “Which measuring device seems to be more accurate?” Why did we use the weaselword “seems”? If one measuring device has a smaller percent error of the mean than another, can’t we say it is more accurate? Unfortunately, the answer is no. To help understand why, imagine that the same student repeated her experiment, only this time instead of using a beaker for the first set of measurements and a graduated cylinder for the second set, she used a single graduated cylinder for both sets of measurements. In this case, would you expect to find exactly the same percent error of the mean for both sets of measurements? Although this is possible, it is extremely unlikely. In fact, if the experiment were repeated many times, chances are that sometimes the first set of measurements would have the lower percent error of the mean and sometimes the second set would. In each experiment, the two sets of measurements, taken with the same graduated cylinder, probably would have different percent errors of the mean because of chance, not because the graduated cylinder is actually becoming more or less accurate from one trial to the next.
As another example, let’s say you toss a coin 100 times and get 59 “heads”. Next, you toss the same coin 100 more times and during the second trial you get only 49 “heads”. Why did you get a different result the second time? One possibility is that the difference was simply caused by chance.  
Heads and Tails 
So, let’s consider the experiment from the previous section where we measured 80 mL of dH_{2}O using a beaker and a graduated cylinder. If the mean volume measured by the beaker were 82.83 mL and the mean volume measured by the graduated cylinder were 80.13 mL, how would we know if the graduated cylinder is closer to the target value because it is more accurate, or if the difference between the two means was simply caused by chance? Unfortunately, there is no way to be sure! No matter how large the difference between the 2 means is, there is no way we can rule out the possibility that it was caused by chance alone. However, what we can say is that the larger the difference between the means, the less likely it was caused by chance alone. In fact, scientists can use statistical tests to determine the probability that we would get a difference as large or larger than the one we observed by chance alone. This probability is called the p value.

The Student’s ttest is a statistical test that can be used to test the null hypothesis that the difference between 2 means was caused by chance alone. By convention, if the p value is greater than 0.05 we conclude that the difference between the 2 means is NOT significant (i.e. there is a relatively high probability that it was caused by chance alone.) On the other hand, if the p value is less than 0.01 we conclude that the difference between the 2 mean is highly significant (i.e. there is a very low probability that it was caused by chance alone.) A p value between 0.01 and 0.05 is considered a borderline region, the difference is considered significant but not highly significant. In this case we would probably want to collect more data before we make a conclusion. 
The ttest is used to decide if 2 data sets come from the same population (Figure A) or from 2 different populations (Figure B). 
IMPORTANT In science, when we accept a hypothesis, this does NOT mean we have decided that the hypothesis is correct or that it is probably correct. It simply means that we do not have convincing evidence to show that the hypothesis is wrong. (Just like in our legal system where we assume someone is innocent until proven guilty, in science we accept our hypothesis until we have convincing evidence to show it is false.) 
Example For example, let’s say we want to test the null hypothesis that when measuring 190 μL of water, there is no difference in the accuracy of the 20 – 200 μL automatic pipettor and the 100 – 1000 μL automatic pipettor. We measure out 190 μL of water 8 times with each instrument and weigh each sample. The mean weight of the water delivered by the 20 – 200 μL automatic pipettor is 0.19 g and the mean weight of water delivered by the 20 – 200 μL automatic pipettor is 0.20 g. The Student’s ttest gives a p value of 0.32. Because p > 0.05 we conclude that the difference in means is not significant. Therefore, we accept our hypothesis that the 2 pipettors are equally accurate. However, this does not mean that we have decided that the 2 pipettors are equally accurate; it simply means that our experiment did not provide sufficient evidence to conclude that they aren’t equally accurate. Of course, it is quite possible that if we repeated our experiment using more trials or using a more accurate scale (one that could read to the nearest 0.001 g or the nearest 0.0001 g) we would find sufficient evidence to show that one pipettor is more accurate than the other. 
Your Turn  
A group of students want to test the hypothesis that when measuring 9 mL of water, a 10 mL graduated cylinder is less accurate than a 10 mL pipette. One student measures 9 mL of water 13 times with a 10 mL graduated cylinder and weighs each sample. The same student repeats the procedure using a 10 mL pipette. The mean weight of the water delivered by the graduated cylinder is 8.89 g and the mean weight of water delivered by the pipette is 9.03 g. The Student’s ttest gives a p value of 0.0062.  
Is the difference between the amounts of water delivered by the 2 measuring devices significant?  Hint  Check your answer. 
Is the difference between the amounts of water delivered by the 2 measuring devices highly significant?  Hint  Check your answer. 
Should they accept or reject their hypothesis?  Hint  Check your answer. 
What should they conclude from their experiment? A. They have proven that their hypothesis is correct. B. Their hypothesis is probably correct. C. The evidence they collected supports their hypothesis D. Their hypothesis is probably wrong. E. They need to collect more data before making any conclusion. 
Hint  Check your answer. 
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