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Power is the probability of rejecting the null hypothesis when it is false

Ergo, power (as its name would suggest) is a good thing; you want more power

A type II error (a bad thing, as its name would suggest) is failing to reject the null hypothesiswhen it's false; the probability of a type II error is usually called

Note Power

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Consider our previous example involving RDI

versus

Then power is

Note that this is a function that depends on the specific value of !

Notice as approaches the power approaches

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We reject if

Under

Under

So we want

Equivalently if-

al pha = 0. 05

z = qnorm( 1 - al pha)pnor m( mu0 + z * si gma/ sqr t ( n) , mean = mua, sd = si gma/ sqr t ( n) , l ower . t ai l = FALSE)

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, , ,

mu0 = 30mua = 32

si gma = 4n = 16z = qnorm( 1 - al pha)pnor m( mu0 + z * si gma/ sqr t ( n) , mean = mu0, sd = si gma/ sqr t ( n) , l ower . t ai l = FALSE)

## [ 1] 0. 05

pnor m( mu0 + z * si gma/ sqr t ( n) , mean = mua, sd = si gma/ sqr t ( n) , l ower . t ai l = FALSE)

## [ 1] 0. 6388

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l i brary ( mani pul at e)mu0 = 30mypl ot

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When testing , notice if power is , then

where

Unknowns: , , ,

Knowns: ,

Specify any 3 of the unknowns and you can solve for the remainder

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The calculation for is similar

For calculate the one sided power using (this is only approximately right, itexcludes the probability of getting a large TS in the opposite direction of the truth)

Power goes up as gets larger

Power of a one sided test is greater than the power of the associated two sided test

Power goes up as gets further away from

Power goes up as goes up

Power doesn't need , and , instead only

The quantity is called the effect size, the difference in the means in standard deviation

units.

Being unit free, it has some hope of interpretability across settings

-

-

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Consider calculating power for a Gossett's test for our example

The power is

Calcuting this requires the non-central t distribution.

power . t . t est does this very well

Omit one of the arguments and it solves for it-

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power . t . t est ( n = 16 , del t a = 2/ 4, sd = 1, t ype = "one. sampl e" , al t = "one. si ded" ) $power

## [ 1] 0. 604

power . t . t est ( n = 16 , del t a = 2, sd = 4, t ype = "one. sampl e" , al t = "one. si ded" ) $power

## [ 1] 0. 604

power . t . t est ( n = 16 , del t a = 100 , sd = 200 , t ype = "one. sampl e" , al t = "one. si ded" ) $power

## [ 1] 0. 604

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power . t . t est ( power = 0. 8 , del t a = 2/ 4, sd = 1, t ype = "one. sampl e" , al t = "one. si ded" ) $n

## [ 1] 26. 14

power . t . t est ( power = 0. 8 , del t a = 2, sd = 4, t ype = "one. sampl e" , al t = "one. si ded" ) $n

## [ 1] 26. 14

power . t . t est ( power = 0. 8 , del t a = 100 , sd = 200 , t ype = "one. sampl e" , al t = "one. si ded" ) $n

## [ 1] 26. 14