Just another random post cause I'm so BORED of studying.
Maths Chapter 18: Normal Distribution
The normal distribution is the most important distribution in statistical theory because it is a suitable model for a very large number of distributions of data. A continuous random variable X is said to have a normal distribution with mean mew and variance sigma square if X has probability function f(x) = whatever, where x is greater or equal to infinity.
Now assuming that I have a box of apples with mean mass 50g which is a large number in the eyes of maths, and standard deviation 20g, based on the graphic calculator, using normalcdf (-E99, 0, 50, 20), I am told that 0.6% of the apples i take out of the box, has a NEGATIVE weight.
Now isn't the normal distribution supposed to be a suitable model since weight is a continuous random variable where there is a probability that the apple's weight can go to inifinity due to variance?
Maths Chapter 19: Sampling, Central Limit Theorem
If X1, X2, ... , Xn are independent random variables, all having the same distribution with mean mew and variance sigma square, then the distribution of the sum, X1+ X2 + ... + Xn is approximately normal, for large values of n, (n>50) whatever the nature of distributions of X.
In other words, when applied to bio, stabalising selection, directional selection and disruptive selection will all result in the same final graph with a normal distribution since all organisms under considering are independent random variables, all have the same distribution and variance (since they are of the same habitat) and have large values (since non-endangered animals surely have a population size of more than 50), whatever the nature of their distribution.
Chapter 20: Hypothesis Testing, Example 3
To test H0: mew = 1.5 ohms against
H1: mew > 1.5 ohms at 0% level of significance (chosen by me since, by definition, if the level of significance is 0%, we will reject H0 when it should not be rejected 0% of the time)
Using GC, bar x = 1.52, s = 0.021213, t value = 2.108, p = 0.0513 > 0
Conclusion: Do not reject Ho and conclude that there is sufficient evidence at 0% level of significance that the wire is pure silver.
Now think about it. at 0% significance, you will reject Ho wrongly 0% of the time. That's better isn't it, so that no mistakes will be made. But wait! if you look at the graph, 0% probability means t value = 0. Meaning, regardless of whether you tested 100% silver 50% silver 1% silver or a ROCK, you will not reject it, because there is NO REJECTION AREA!!
My conclusion is that we are learning USELESS THINGS.
Ok that's enough. Back to learning all the nonsence
Thursday, November 6, 2008
Exam thoughts
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3 comments:
Hi Jack,
Would like to gently correct some of the conceptual errors you've shown in the post :)
1) It is not possible for a box of apples to have standard deviation 20g.
Standard deviation is a bit misleading, variance is actually a better measure of variation.
Variance is equal to standard deviation squared. This means that the variance is 400 g!
Now how do we compute variance? Well it's simply the sum of the squares of the individual masses of apples minus the mean mass.
If the variance is 400g, you potentially have apples of negative mass, which is exactly what you showed to be impossible.
But in the first place, the standard deviation cannot be 20g! So there you go..
2) What you've stated is the Central Limit Theorem, which relates to a SUM of independent random variables that are identical.
But what you've stated is the INDIVIDUAL distributions of random variables, that are potentially different.
So, your counterexample doesn't apply.
Besides, the Central Limit theorem provides a good approximation. Not exactly the same.
3) You cannot pick your significance level to be 0%. That is absurd! Saying that the significance level is 0% is as good as saying you are sure something is that thing.
So you've got yourself in a circular argument. You claim that you can pick the significance level at 0% (which means you are SURE it is something), and then you pick something that is NOT it, and show that you CANNOT reject it.
But in the first place, if you want to do a statistical test, you must be UNSURE about what it is. That's why you take a sample and do a statistical test.
Then again, that's Statistics, not Mathematics. Statistics is as different from Mathematics as Medicine is different from Life Sciences.
God bless,
Jonas
O.O we got a math professor on my blog!! Haha hi Mr Chow how did you find my blog XD
havn't been here for a while. haha well I found it through Fabriz's blog..
have a good break and hope you did well for math!
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