Central Limit Theorem

The random variable of interest as X(hyphen) represents the mean of a sample of n Observation from the raw observation X, as in x1,x2,x3 …,xn. We call it an actual finite mean <meau x< finite. finite SD 0<sigma x<infite.It will help us find that the probability distribution of the same mean varies depending on the SD’s value.

When the true value of SD sigma x is known, then X is normal, sampling distribution of X hype is a normal distribution with mean Meauc and SE sigma x/sqrt( n)

If X is not normal, the sampling distribution of X hype is aprroxmately normal , with mean Meauc and SE sigma x/root n, and this approximation improves as n->finite, and this is known as Central limit Therom Support that the daily maximum temperature in the month of January in Dunedin, NZ, follows the normal distribution with a mean of 22 degresss celsius and a standard decviation of 1.5 degrees., Then in line with a comments for situation 1 for samples of size n=5, the samplling distribution of Xhypen will be normal.

knitr::opts_chunk$set(echo=FALSE)
## [1] 0.6708204

Chypen is aclearly a taller, skinner normal distribution

Asking other probability questions.distinguishing between the measurement distribution and the sampling distribution is important. P(X<21.5) randomly chosen day in Jan has maximum temperature of less than 22.5 Then based on 5 random days

## [1] 0.3694413
## [1] 0.2280283
## [1] 0.2280283
## Error in xy.coords(x, y, xlabel, ylabel, log): 'x' and 'y' lengths differ
## Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet
## Error in eval(expr, envir, enclos): object 'fx.samp' not found
## Error in segments(lx1, ly1, lx2, ly2, ...): plot.new has not been called yet
## Error in cbind(c(21.5, xvals.sub), c(0, fx.samp.sub)): object 'fx.samp.sub' not found
## [1] 20.94998 24.67142 22.05466 22.11257 22.18445

Above line code - 5 randomlygrnerated temperature freom the X–N(22,1.5)

if we dont have meaux(22) and sigmax( 1.5). we must use the xhypen and s

## [1] 22.39462
## [1] 1.370229

The estimated standard error calculation

## [1] 0.6127851

sampling distribution of Xhypen using the t distribution with 4 degree of freedom

## [1] -1.459918

if we using the estimate s , t4 follows the the estimated probability is

## [1] 0.1090454

Interval for a mean.Suppose you know the observations are normally distributed but dont know the tru mean meaux or tru standard deviations sigma x

## [1] 21.28634 21.41585 21.49237 20.49837 22.42467

finding sample mean in its sampling distribution. First calculate the sample mean xhypen .the standard deviaton s

## [1] 21.42352
## [1] 0.6858966
## [1] 0.3067423

lets say 95% CI for the tru, unknown mean meaux. So alpha=0.05 .t distribution with n-1=4 degree of freedom.For a central area of 0.95 unser this curver alpha/2

## [1] 0.975
## [1] 2.776445
## [1] 0.95
## [1] 20.57187
## [1] 22.27517

Alter the intervals to change the cofidence level Chnage the critical value.much define alpha/2 in each tail. For example 80% percent CI(alpha=0.2) and 99 % CI(alpha=0.01)

## [1] 20.95322 21.89382
## [1] 20.01125 22.83579