When investigating a probability distribution of some popula

Solution

Let me explain with an example!

Say, x1,x2,....,xn be random variables following a normal distrbtn with mean=mu and s.d known...

Then for a given sample of size n...we know that sample mean = x-bar...and sample s.d is known ...so.we have to estiamte popltn s.d by sample s.d.....
so, x~N( mu, popltn s.d)......so,x-bar ~ N ( mu, popltn s.d/ sqrt(n) )
So, sqrt(n) * ( x-bar - mu ) / popltn s.d ~ N ( 0,1)...right?...
now, we don\'t know popltn s.d...but it is in the equation...so,we have eliminate this term..so that we can calculate it..right?
Now, (n-1) (sample s.d^2) / (popltn s.d)^ 2 follows a chi-sq distrbtn with d.f n-1..
and both of the distrtns here are independent..
so, our objective is to eliminate the unknown popltn s.d from the test stat...
so, we have to sqrt the 2nd term and we have to divide the 1st term by 2nd term..

i.e, [sqrt(n) * ( x-bar - mu ) / popltn s.d ] / [sqrt [ (n-1) (sample s.d^2) / (popltn s.d)^ 2 ] ]............

= sqrt(n) * ( x-bar - mu ) /  [sqrt [ (n-1) (sample s.d^2) ] ] ...here we eliminate the popltn s.d term...
all the term is known to me..but this does not follow any specific known distribution...so, how can we calculate prob. from it when we don\'t know it\'s distrbtn?
Now, from the deftn of T-distrbtn,
Normal / sqrt [ ( chi -sq) / (n -1) ] follows a t-distbtn with (n-1) df..
so, we divide our test stat.\'s denominator by by sqrt(n-1) and we get a knwon distrtn..
so, hence test stat is= sqrt(n) * ( x-bar - mu ) / (sample s.d)

and when we are given a certain level of confidence...we check what should be the range of the test statistic to accept it...and compare it with our calculated test statistic..if the test stat falls withing the range..then we accept it..and from knowing the distribution..we can calculate it\'s prob..any kind of prob. from it..