Hey guys please help your friend on this problem The solutio

Hey guys please help your friend on this problem. The solution is attached. For part c) I dont get why my professor use that formula and how come he reduces down to mean t = mean (1-s)/s and also for standard deviation to s/s ? Can someone show me step by step how from the first equation to the last equation? I am new learner and I have hard time with statistics. Deeply appreciated.

Solution

C.

Part c is looking for the mean of all values of (x/s - Xbar).

Thus,

Mean (x/s - Xbar)

Note that you can separate these into two means,

= Mean (x/s) - Mean (Xbar)

As s is just a constant, the Mean(x/s) = (1/s)Mean(x).

Also, as Xbar is a constant, then Mean(Xbar) = Xbar. Thus,

= (1/s) Mean(x) - Xbar

As Mean(x) = Xbar,

= (1/s) Xbar - Xbar

Factoring Xbar,

= Xbar (1/s - 1)

or, combining the terms inside the parenthesis using just one fraction,

Mean (x/s - Xbar) = Xbar (1 - s) / s [the mean of the new quantity]

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Now, the second part wants the standard deviation of (x/s - Xbar)

s(x/s - Xbar)

Separating the two terms,

= s(x/s) - s(Xbar)

As s is just a constant, we can take it out, that is, s(x/s) = (1/s) s(x).

As Xbar is a constant, it has no standard deviation, that is, s(Xbar) = 0. Thus,

= (1/s) s(x) - 0

As s(x) = s, then

= (1/s) (s)

= 1. [the standard deviation of the new quantity]

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