Question

6. (1 pt) There are an infinite number of possible normal distributions and we use the formula z =(x - m)/ s where m = mean and s = standard deviation, to convert values from any normal distribution to the standard normal distribution. We then calculate the answer in terms of the standard normal distribution and say that the answer we got is also the answer for the normal distribution we were working with. Why are we allowed to do that? (i.e why are we allowed to say that our answer that holds for the standard normal distribution also holds for any normal distribution?). (Hint; think about the formula for z).

Answer 1

There are an infinite number of possible normal distributions and we use the formula z =(x - m)/ s where m = mean and s = standard deviation, to convert values from any normal distribution to the standard normal distribution.

Here Z follows standard normal distribution i.e. with mean 0 & std deviation 1.

& x follows the normal distribution with mean m & std deviation s

by using Z = ( x-m) /s

we have converted normal distribution to std normal distribution.

& found the value for the probability.

So we can reverse the procedure and use

z =(x-m) /s

x-m =z*s

x = m + z*s

We can convert std normal distribution to a normal distribution.

& corresponding proabbiity will belong to x.

So we say that our answer that holds for the standard normal distribution also holds for any normal distribution.

Because using x = m + z*s, we can convert std normal to any normal distribution.