# determine the critical region and test statistic that would be used to test the null hypothesis at the `(1-alpha)`% level of significance`H_0: mu<=82`, `H_A: mu>82` st dev 0.10 `H_0:...

determine the critical region and test statistic that would be used to test the null hypothesis at the `(1-alpha)`% level of significance

`H_0: mu<=82`, `H_A: mu>82` st dev 0.10

`H_0: mu=44`, `H_A: mu != 44` st dev 0.01

`H_0: mu >=17`, `H_A: mu < 17` st dev 0.05

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### 1 Answer

Call the significance level `(1-alpha)`%

Let ` z_(1-alpha)` be the `(1-alpha)`th percentile of the standard Normal distribution

a) The critical region to detect `mu>0` where `Z` ~ N`(mu,1)` is given by

`z: z in [z_(1-alpha),oo)`

Convert the value of `x`to a z-score` `:

Test statistic = `(x-82)/0.1`

The critical regi` `on for `x` is given by

`x: (x-82)/0.1 in [z_(1-alpha),oo)`

ie `x: x in [0.1z_(1-alpha) + 82,oo)`

b) The critical region to detect `mu != 0` where `Z` ~ N`(mu,1)`

is given by

`z: |z| in [z_(1-alpha/2),oo)`

Convert the value of `x` to a z-score:

Test statistic = `(x-44)/0.01`

The critical region for `x` is given by

`x: |(x-44)/0.01| in [z_(1-alpha/2),oo)`

ie `x: x in [0.01z_(1-alpha/2) + 44,oo)` or `x: x in [44 - 0.01z_(1-alpha/2),-oo)`

c) The critical region to detect `mu<0` where `Z` ~ N`(mu,1)`

is given by

`z: z in [-z_(1-alpha),-oo)`

Convert the value of `x` to a z-score:

Test statistic = ` (x-17)/0.05`

The critical region for `x`is given by

`x: (x-17)/0.05 in [-z_(1-alpha),-oo)`

ie `x: x in [17 - 0.05z_(1-alpha),-oo)`

**a) test statistic (x-82)/0.1**

** critical region [z(1-alpha),infinity)**

**b) test statistic (x-44)/0.01**

** critical region [z(1-alpha/2),infinity) or [-z(1-alpha/2),-infinity)**

**c) test statistic (x-17)/0.05**

** critical region [-z(1-alpha),-infinity)**