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Imagine that the human size is influenced by a gene with two different alleles (gene variants) $A$ and $a$. Concerning this topic you have collected the following data: $\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \text{proband} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\ \hline \text{number of A-lleles} & 0 & 2 & 2 & 1 & 0 & 0 & 1 & 2 & 1 & 2\\ \text{size (cm)} & 159 & 182 & 184 & 168 & 162 & 161 & 165 & 195 & 167 & 186\\ \end{array}$ For each proband you write down how many A-alleles the proband has. Determine the strength of the connection by calculating the regression coefficient of a linear regression. To do so, assume that the human size is normal distributed with the same variance. After that have a thought on the following question: Why is this task problematic?

Hey, I did the calculation with R, it is rather short. The code is

rm(list=ls(all=TRUE))}

x <- c(0,2,2,1,0,0,1,2,1,2)

y <- c(159,182,184,168,162,161,165,195,167,186)

print(cor(x,y))

My result is that the regression coefficient is $$ r_{XY}=0.9236728. $$

This should be right. And this result seems to show that there is a rather strong linear connection between the number of A-alleles and the human size. Anyhow this result resp. the task seems to be problematic.

Why?

In my opinion because of two points: The assumption that the variance is the same is problematic, because f.e. this makes no difference between male and female, although there probably is a much more bigger variance in male human size than in female human size.

On the other hand the assumption of normal distribution does not forbid negative human sizes which are impossible.

Am I right or why is the task problematic?

mathfemi
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1 Answers1

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Your first comment concerning male vs female could be very relevant, however you would need some data to verify this. The second remark is not so relevant since the tail of this normal approximation that goes under zero is very very very small.

I think that your teacher wants to hear something about the data you have. You could think about whether the amount of data you have is enough to reach strong conclusions and what the effect could be of the fact that you have very litlle dispersion in your $x$ values ie the number of aleles.

Marc
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