Question

Law of Large Numbers, Central Limit Theorem, and Confidence Intervals 1. (15 points) In an exercise, your Professor generated random numbers in Excel. The mean is supposed to be 0.5 because the numbers are supposed to be spread at randonm between 0 and 1. I asked the software to generate samples of 100 random numbers repeatedly. Here are the sample means x for 50 samples of size 100: 0.532 0.450 0.481 0.508 0.510 0.530 0.4990.4610.5430.490 0.497 0.5520.473 0.425 0.4490.507 0.472 0.438 0.527 0.536 0.492 0.484 0.498 0.536 0.492 0.483 0.529 0.490 0.548 0.439 0.473 0.516 0.5340.540 0.525 0.540 0.464 0.507 0.483 0.436 0.4970.4930.458 0.527 0.4580.510.4980.480 0.479 0.499 The sampling distribution of x is the distribution of the means from all possible samples. We actually have the means from 50 samples. (a) Make a histogram of these 50 observations (b) Write down the Law of Large Numbers and the Central Limit Theorem, in your own words (c) Compute the mean and the median of the sample. Use this information and the histogram in (a). Does the distribution appear to be roughly normal, as the central limit theorem says will happen for large enough samples?

Answer 1

###By using R command

> x=c(0.532,0.450,0.481,0.508,0.510,0.530,0.499,0.461,0.543,0.490,0.497,0.552,0.473,0.425,0.449,0.507,0.472,0.438,0.527,0.536,0.492,0.484,0.498,0.536,0.492,0.483,0.529,0.490,0.548,0.439,0.473,0.516,0.534,0.540,0.525,0.540,0.464,0.507,0.483,0.436,0.497,0.493,0.458,0.527,0.458,0.510,0.498,0.480,0.479,0.499)
> x
[1] 0.532 0.450 0.481 0.508 0.510 0.530 0.499 0.461 0.543 0.490 0.497 0.552
[13] 0.473 0.425 0.449 0.507 0.472 0.438 0.527 0.536 0.492 0.484 0.498 0.536
[25] 0.492 0.483 0.529 0.490 0.548 0.439 0.473 0.516 0.534 0.540 0.525 0.540
[37] 0.464 0.507 0.483 0.436 0.497 0.493 0.458 0.527 0.458 0.510 0.498 0.480
[49] 0.479 0.499
> hist(x)
> summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.4250 0.4745 0.4970 0.4958 0.5265 0.5520

a)

Histogram of x 12 12 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56

b) According to the Weak law of Large Number

sample Moment converges in law to population Moments

Xbar -------> mu (Population mean)

According to the central limit theorem as sample size (n) becomes sufficiently large the sampling distribution of sample mean converge to Normal distribution.

that is

sqrt(n)*((Xbar-mu)/sigma) ----------> N(0,1) (convergence in distribution)

c) From the Above Histogram the distribution of the data is roughly normal.