I don't think heights are actually normally distributed. For example, there's probably a bump for "little people" (formerly called midgets) that has no corresponding "big people" bump. I think a "normal" drawfism height spans to something below 3 feet, and if the average height for an adult is at least 5 feet tall (which I think is correct), we would expect to see "normal tall people" of 8 feet and taller, but we don't.

The central limit theorem doesn't say that everything is normally distributed, which is a common error of interpreting it. However, the normal distribution is at least a reasonable approximation. If you're looking for a physical mechanism, it's probably something like there being a confluence of a large number of random factors which, when combined, happen to give something that looks approximately normal.

You might also look up "fat tails" in financial jargon, which correspond to events that are rare but more common than what a normal distribution would predict.

I don't see how the CLT applies. I agree that if we were talking about the average height of 100 people, that would be normally distributed after alot of samples even if human height followed a (say) uniform distribution. But the question seems to be why is the underlying distribution of human height (presumably split by sex) normal.

Because height is multifactorial?

If it was just one factor there would be only (equally) short and (equally) tall people.

CLT doesn't answer the question of "why is height normally distributed". I think it answers

I agree that I don't know why or if height is normally distributed.

CLT says that sampling distribution of the mean of any population will be approximately normal if the sample is of sufficient size. We approximate height as normal (conditional on gender because once you combine men and women you have a bimodal distribution that isn't normal) because the normal is a really flexible distribution where you can use the parameters to fix the mean and variance. Clearly height cannot actually be normal because that would imply that it is possible to have someone with an negative height (which would be many sigmas below the mean but still have a non-zero density)

I guess I'm still not following. What is the "physical property" that accounts for height (and other things) being normally distributed?

.

I was thinking that the CLT helps explain the somewhat surprising ubiquity and usefulness of the normal distribution. So the answer to the question "is there a physical property that things with normal distributions have in common" would be "not really", or perhaps more precisely ¯\_(ツ)_/¯

Fair enough. Height might be normally distributed due to the CLT if an individuals height can be thought of as randomly sampling of some space of protein expression or whatever, but I don't know enough biology to determine if that sentence even makes sense. (I imagine it doesn't)

Anyone know how to do this? By decider is meant a turing machine that halts.

Would appreciate some insight/solution to this problem:



The question is - What is the minimum coefficient of static friction between the ground and ladder such that it won't slip?

The rope is attached to the ceiling (it's creating a tension, and holding the ladder in that position). Btw, it looks like the angle b/w the rope and ladder is 90 degrees, ignore that.

I got an answer, but I'm pretty sure I'm wrong. Would like to see a solution.

these physics questions like this you can usually google and find an answer / explanation on how to do it via yahoo answers (or on chegg if you have a paid account there)

I have a regression related question.

There exist two types of cars- Ford and Budget.

Aim: I want to establish whether there's a difference in the rate of depreciation between the two cars.

My model takes the form:

Log (Price_i) = B0 + B1.Age_i + B2.Dummy_i + B3.Age_i.Dummy_i + e_i

where,
Price_i=price of car i
Age_i=age of car i
Dummy_i=a dummy variable that takes the value 1 if car i is a Ford and 0 otherwise

Intuitive explanation
Cars in the early stage of life depreciate faster than their older counterparts- think a brand new car depreciating at 30% a year vs the same car but at 15 years old, perhaps only depreciating at 10% a year. A variable for age (Age_i) is thus included in my model to ensure that the aim of my study isn't undermined by mere differences in age between the two groups (Ford and Budget).

Now let's say i obtain the following results:

B1= a significant p-value (which, consistent with my intuitive explanation, is to be expected)
B2= a significant p-value
B3= a non-significant p-value

Now, here's the point of my confusion. I've been informed that B3, not B2, addresses my aim (bolded above).
My question is why? After all, any differences in age between the two groups (Ford and Budget) are surely caught by the Age variable (B1.Age_i), so why can't we just interpret B2 and perhaps not even include the interaction term in the model? Why would doing this be wrong?

Here is my take. You are not thinking about this quite the right way. Suppose the world is such that Fords cost $25,000 when new and Budgets cost $10,000 when new. And each car loses $1,000 in value each year (for this simple explanation I am using absolute depreciation rather than percents, but the point is the same).

In this world, to fully capture the price of a vehicle you would need one variable for its age and another (dummy) variable for whether it is a Ford or Budget. You essentially need to capture two pieces of information.

Now, complicate the world a bit. Suppose that Fords lose $2,000 in value each year and Budgets lose $1,000 in value each year. In this world, to fully capture the price of a vehicle, you need to capture three pieces of information (differential in new prices between Ford and Budget, and how much each depreciates per year).

In terms of setting up dummy variables, this can be accomplished by having one variable for its age, one dummy variable for whether it is a Ford or Budget, and another dummy variable for an "interaction" effect (that is, the differential between the two depreciation values).

You may want to create some hypothetical datasets and run simple regressions on them to see what is going on in these dummy variable (with and without interaction effects) regressions.

short answer: b2 is the constant difference in value for budget cars and fords, b3 is the difference in depreciation rates.

The easiest way to see this is to think about some fitted prices. The predicted value of a budget car is

b0 + b1*age

because b2 and b3 are both multiplied by the dummy which is 0. This is just a line. The predicted value of a for is

b0 + b1*age + b2 + b3*age because the dummy is equal to 1 so it doesn't alter the b2 or b3 terms. We can rearrange this to be

(b0 + b2) + (b1 + b3)*age

Again price is a linear function of age. For both of these lines there is some initial value to the car that decreases by a constant amount for each year of age (the depreciation rate). You can see pretty clearly from these equations that b2 is part of the initial value difference in ford and budget cars while b3 is changing the depreciation rate between these two types of cars.

One additional comment prompted by the above excellent reply.

If you read OP's description, there appears to be confusion about the underlying "model" which is manifest by a misunderstanding of the dummy variables. (For example, why an Age dummy variable is needed in the regression.) I think in these cases it is often important to clarify the "modeling" rather than focus upon the regression's variables.

Too many students are taught to believe that regressions are equivalent to the underlying model. We have a dataset, let's run some regressions and see what we can find. There is an important difference between a "reduced form" and a "structural model" which is a serious issue in statistics and econometrics.

Statistical modeling should first and foremost be modeling. Too often we lose sight of that.

Thanks very much to you both. I believe i get it now.

Let's say I changed my aim, as absurd as it may sound, to the following:

Aim: I want to know which car (Ford or Budget) has a higher starting price.

Then I wouldn't bother with an interaction term right? My model would simply be,

Log (Price_i) = B0 + B1.Age_i + B2.Dummy_i + e_i

And the B2 variable would reliably expose which of the brands have a higher starting price right?

Now b2 is going to be the difference in price between Fords and budget cars at every age. Going back to fitted values, a budget car price will be

b0 + b1*age

while a ford price will be
(b0 + b2) + b1*age

so b2 is the price difference which will be constant with respect to age. This time instead of fitting 2 lines with different intercepts and different slopes you are fitting 2 lines with different intercepts but imposing the structure that they have to have the same slope. This may or may not be a correct thing to do based on your data.

Hello,

I am having some trouble finding the value of x and the weighted mean from this table.

For three subjects: French; German and English; the weighted mean is: 2; x; and 3 respectively.

Jack's results are: 80; 72 and 46 respectively.
Jill's are 64; 82 and 40 respectively.

I tried to express both as simultaneous equations but I am sill left with an unknown m and mx value that I cannot break down further. Could I ask someone how I should approach this problem correctly?

Thanks in advance.

I thought more about that regression question and whosnext is right. OP needs to think about the modeling decisions. Even if the quantity of interest is the initial price difference you still probably want to use the interactive model. You can fit any regression you want but that doesn't make it right. In this case given the very reasonable belief that the cars depreciate at different rates (which may make the slower depreciating cars more initially valuable) you want to let the slopes vary. Without the interaction you are finding two parallel lines that have a slope equal to the average of the two depreciation rates.

So if I understand it correctly 2, x and 3 are the weighted means of the results of Jack and Jill for French, German and English respectively? This would be a bit weird because the sum of the weights doesn't add up to one in that case.

What are these m and mx values you mention, are these the weights? If I understand the question correctly you should be able to solve the equations for French and English for the weights.

Sorry, my wording wasn't clear.

The results are percentages so when they are divided back down by the weights they should sum to less than one.

M is the mean.

I wrote two simultaneous equations.

For Jack:
(1) 298+72x/5+x= mean

For Jill:
(2) 248+82x/5+x= mean

I tried to solve these by multiplying the equations by (5+x) and 82 and 72 respectively. This still leaves me with an M that I cannot solve for.

It's still not really clear to me. So for French, Jack got an 80 and Jill got a 64 and the weighted mean of these two scores is 2. Is that correct?

It sounds like what you intend is :

(298 + 72x)/(5 + x) = mean

(248 + 82x)/(5 + x) = mean

assuming you intend that the mean for Jack and the mean for Jill are to be the same, then

298 + 72x = 248 + 82x

so x = 5 and mean = 65.8

If the means are not the same, then you have two equations for three unknowns.

I think he means the weights are 2/(5+x), x/(5+x), and 3/(5+x).