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Example: Analyzing the difference in distributions

Finding the probability that a randomly selected woman is taller than a randomly selected man by understanding the distribution of the difference of normally distributed variables.

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• why do we use man minus woman?and not woman minus man?
• That is a great question.
In the video we have D = M - W and the Z-score of -0.8 gives P(D < 0) = 0.2119 under the N(8, 100) distribution.

If we define D = W - M our distribution is now N(-8, 100) and we would want P(D > 0) to answer the question. Our Z-score would then be 0.8 and P(D > 0) = 1 - 0.7881 = 0.2119, which is same as our original result.

The difference between the approaches is which side of the curve you are trying to take the Z-score for.
• I don't understand why did we set the probability P(D<0)?
• D is the difference between men and women's height: M-W.
If D is less than 0 ie. negative than that would mean the woman was taller than the man. Man 170 cm Women 175 cm D = M-W D = 170-175 D = -5.
• I went ahead and solved this before watching the rest of the video.
My logic was slightly different and I ended up getting a Z-score of 0.8 instead of -0.8 which resulted in a slightly different value for the Probability (got 0.2119 instead of 0.212).

I'd like to know why my way of thinking about this problem is wrong.
For my Z-score I did 178-170 (X - mean) because my mean is the women's mean 170 and for all the women taller than the men they must have at least a height surpassing the men's mean 178. Then, I got a positive 8 then divide by 10 (sum of the 2 variances). Hence, I got a positive 0.8 Z-score.

I got the result mentioned above because when I drew out normal distribution on paper I drew out the men's then the women's on top of it because the -1SD of men (170) is the women's mean, instead of drawing out 3 different normal distribution graphs like Sal did.

When drawing this way, it led me into thinking that I needed to find the area under the curve of the women's distribution on the right from 178 towards positive infinity. Hence, I did (178-170)/10 to get a positive 0.8 Z-score. :(

But when Sal showed that D = M-W and to find the women taller than men P(D<0) this one makes sense to me as well. I find his way makes sense when he draws out 3 different graphs and my way makes sense when I superimpose 2 distribution graphs. So, I guess my mistake here was that I superimposed the graphs and for future solving of this kind of problem I should never superimpose to avoid being misled?
• You just had a different view of the problem, which is also correct. The difference is just that he rounded his answer. If you look at the calculator output, it matches your answer.
(1 vote)
• Sorry what if we don't have TI 84 calculator tool? how do you do it manually? without calculator ? thanks
• You can do it manually by consulting a z-score table (e.g., http://users.stat.ufl.edu/~athienit/Tables/Ztable.pdf). You find the cross section that matches your z-score, and it will give you what his calculator just did.
(1 vote)
• Is there more than this, because my AP stats class already past this stuff in the 1st quarter?
(1 vote)
• Why do we want to find the difference? (M - W) instead of find the sum of each independent variables? (M + W)
(1 vote)
• We're interested in the difference in height between the man and the woman, not their combined height. Subtracting the heights allows us to directly compare their heights to determine which one is taller.
• For combining for normally distributed variables, the independence clause needs to be there? Since variance is the spread of the data.
• 1. for mean, no need. you can just add or subtract each of means

2. but for variance and standard deviation, we have to consider the independence between them. cause there's a concept of "co"variance when they are dependent on one another, which means some parts of their values vary together. you can see it as a kind of amplified wave when two similar ones meet together

3. for example
X = how many salty snacks i ate
Y = how much water i drank with them
the more varied X is, the more varied Y would be, we can guess
and vice versa
so var(X+Y) = var(X) + var(Y) + covar(X,Y)
#covar(X,Y) tells how much more would you consume X or Y by the consumption of Y or X

one more thing, what about var(X-Y)? would you think you still have to add covar(X,Y) to var(X)+var(Y)? or subtract? or need a whole new formula? please sleep on it
• Wouldn't -1E9999 work better or even -1E9999999999999999999999999
You know what I mean right?
(1 vote)
• Yes, using a very large negative number like −1E9999 or −1E9999999999999999999999999 would also work to approximate negative infinity in the context of a calculator. This ensures that we're capturing the entire left tail of the distribution when calculating P(D < 0).
(1 vote)
• Can someone help me understand why we subtract the random variables in order to find the answer to the question?
(1 vote)
• We subtract the random variables (M and W) to find the difference in height between a randomly selected man and a randomly selected woman. This difference tells us whether the woman is taller, shorter, or the same height as the man.
(1 vote)
• why the difference of variance is the sum of two variables but not subtract them?
(1 vote)