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let's say that you're solving a problem

associated with hypothesis testing and

let's say that the null hypothesis is

that the mean is 50 and that for the

alternative hypothesis let's say someone

believes that the mean is not 50 and

decides to conduct a test how do you

know if the null hypothesis should be

rejected or not rejected for this

example we need a two-tailed test the

area shaded in blue is known as the

rejection region the area that is not

shaded is the fail to reject region now

the Z values that separate these two

regions these are the critical values

now let's say if we're dealing with a

95% confidence level you could use the

standard Z table to get the Z values so

at a 95% confidence level the critical

values will be 1.96 on both sides now in

order to know whether you should accept

or reject the null hypothesis you need

to get another Z value now calculate a Z

value and compare it to a critical value

let's call that calculate a Z value ZC

let's say if ZC is to the right of Z

then that means that you should reject

the null hypothesis if ZC is to the left

of Z that is if it's in the fail to

reject region you shouldn't reject the

null hypothesis you should keep it

so now let's talk about how we can

calculate zc so this is known as the

test statistic so first so let's talk

about if we have a population mean

versus a population proportion now

sometimes you may need to use the

t-distribution other times we need to

use the normal distribution and get the

z value so let's say that the sample

size is less than 30 and that the

population standard deviation is unknown

so if these two conditions are met in

that case we need to use the T

distribution so our T value is going to

be the sample mean minus the population

mean divided by the sample standard

deviation over the square root of n

where n is the sample size now let's say

that n is less than 30 but we know the

population standard deviation in this

case we could use a normal distribution

so we're going to calculate the z value

so our calculated z value is going to be

the sample mean minus the population

mean divided by the standard deviation

that is the population standard

deviation over the square root of n now

let's say that n is greater than 30 and

the population standard deviation is

unknown because the sample size is large

the distribution will be similar to a

normal distribution so once again we can

calculate the Z value so it's going to

be the sample mean minus the population

mean divided by the sample standard

deviation over the square root of n now

if n is large

and if we know the population standard

deviation everything is going to be the

same

instead of using s we're going to use

Sigma so those are the formulas in which

we can calculate the Z value or our T

value and compare it to the critical

value in order to determine whether we

should reject the null hypothesis or if

we should not reject it so this is when

you have when you're dealing with the

population mean now sometimes you can be

dealing with the population proportion

and so that's going to be P instead of

mu in this case we're going to use a

different formula to calculate the Z and

so the formula is it's going to be the

sample proportion minus the population

proportion divided by the square root of

P Q over N now n is still inside the

square root q is 1 minus P so just keep

that in mind so that's the formula we

need to use if we're dealing with a

proportion that's how we can get our

calculated Z value so those are the test

statistics that we need to be using when

solving problems associated with

hypothesis testing