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# Example: Analyzing the difference in distributions

AP.STATS:
VAR‑5 (EU)
,
VAR‑5.E (LO)
,
VAR‑5.E.1 (EK)
,
VAR‑5.E.3 (EK)

## Video transcript

suppose that men have a mean height of 178 centimeters with a standard deviation of 8 centimeters women have a mean height of 170 centimeters with a standard deviation of 6 centimeters the male and female heights are each normally distributed we independently randomly select a man and a woman what is the probability that the woman is taller than the man so I encourage you to pause this video and think through it and I'll give you a hint what if we were to define the random variable M as equal to the height of a randomly selected man height of random man what if we define the random variable W to be equal to the height of a random woman woman and we defined a third random variable in terms of these first two so let me call this D for difference and it is equal to the difference in height between a randomly selected man and a randomly selected woman so d the random variable D is equal to the random variable M minus the random variable W so the first two are clearly normally distributed they tell us that right over here the male and female Heights are each normally distributed and we also know or you're about to know that the difference of random variables that are each normally distributed is also going to be normally distributed so given this can you think about how to tackle this question the probability that the woman is taller than the man all right now let's work through this together and to help us visualize I'll draw the normal distribution curves for these three random variables so this first one is for the variable M and so right here in the middle that is the mean of M and we know that this is going to be equal to 178 centimeters we'll assume everything is in centimeters we also know that it has a standard deviation of 8 centimeters so for example this is one standard deviation above this is one standard deviation below this point right over here would be 8 centimeters more than 178 so that would be 186 and this would be 8 centimeters below that so this would be 170 centimeters so this is for the random variable M now let's think about the random variable W the random variable W the mean of W they tell us is 170 and one standard deviation above the mean is going to be 6 centimeters above the mean the standard deviation is 6 6 centimeters so this would be minus 6 is to go to one standard deviation below the mean now let's think about the difference between the two the random variable D so let me think about this a little bit the random variable D the mean of D is going to be equal to the differences in the means of these random variables so it's going to be equal to the mean of M the mean of M minus the mean of W minus the mean of W well we know both of these this is going to be 178 minus 170 so let me write that down this is equal to 178 centimeters minus 170 centimeters which is going to be equal to I'll do it in this color this is going to be equal to 8 centimeters so this is 8 right over here now what about the standard deviation assuming these two random variables are independent and they tell us that we are independently randomly selecting a man and a woman the height of the man shouldn't affect the height of the woman or vice versa assuming that these two are independent variables if you take the sum or the difference of these then the spread will increase but you won't just add the standard deviations what you would actually do is say the variance of the difference is going to be the sum of these two variances so let me write that down so I could write vary ins with var or I could write it as a standard deviation squared so let me write that the standard deviation of D of our difference squared which is the variance is going to be equal to the variance of our variable M plus the variance of our variable W now this might be a little bit counterintuitive this might have made sense to you if this was plus right over here but it doesn't matter if we are adding or subtracting and these are truly independent variables then regardless of whether we're adding or subtracting you would add the variances and so we can figure this out this is going to be equal to the standard deviation of variable m is 8 so 8 squared is going to be 64 and then we have 6 squared this right over here is 6 6 squared is going to be 36 you add these two together this is going to be equal to 100 and so the variance of this distribution right over here is going to be equal to 100 well what's the standard deviation of that distribution what's going to be equal to the square root of the variance so the square root of 100 which is equal to 10 so for example one standard deviation above the mean is going to be 18 one standard deviation below the mean is going to be equal to negative 2 and so now using this distribution we can actually answer this question what is the probability that the woman is taller than the man well we can rewrite that question is saying what is the probability that the random variable D is what conditions would it be pause the video and think about it well the situations where the woman is taller than the man if the woman is taller than the man then this is going to be a negative value then D is going to be less than zero so what we really want to do is figure out the probability that D is less than zero and so what we want to do if we say zero is right over if we said that zero is right over here on our distribution so that is d is equal to zero we want to figure out well what is the area under the curve less than that so we want to figure out this entire area there's a couple of ways you could do this you could figure out the z-score for D equaling zero and that's pretty straightforward you could just say this Z is equal to zero minus our mean of 8 divided by our standard deviation of 10 so it's negative 8 over 10 which is equal to negative 8/10 so you could look up a Z table and say what is the total area under the curve below Z is equal to negative 0.8 another way you could do this is you could use a graphing calculator I have a ti-84 here where you have a normal cumulative distribution function I'm going to press 2nd vars and that gets me to distribution and so I have these various functions I want normal cumulative distribution function so that is choice two and then the lower bound well I want to go to negative infinity well calculators don't have a negative infinity button but you could put in a very very very very negative number that for our birth business is equivalent to negative infinity so we could say negative 1 times 10 to the 99th power and the way we do that is second this to capital ease or saying essentially times 10 to the and I'll say 99 power so this is a very very very negative number the upper bound here we want to go let me delete this the upper bound is going to be 0 we're finding the area from negative infinity all the way to 0 the mean here well we've already figured that out the mean is 8 and then the standard deviation here we figured this out too this is equal to 10 and so when we pick this we're going to go back to the main screen enter so this is we could have just typed this in directly on the main screen this says look we're looking at a normal distribution we want to find the cumulative area between two bounds in this case is from negative infinity to 0 from negative infinity to 0 where the mean is 8 and the standard deviation is 10 we press enter and we get approximately zero point two one two is approximately zero point two one two or you could say what is the probability that the woman is taller than the man well zero point two one two or approximately there's a twenty one point two percent chance of that happening a little better than one in five
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