A hypothesis will be used to test that a population mean equals 6 against the alternative that the population mean does not equal 6 with known variance σ. What are the critical values for the test statistic Zo for the significance level of 0.06?

Both μ and [tex]σ^{2}[/tex] are parameters of a normal distribution. So if we know the value of σ, we can easily know the value of [tex][tex]Z=\frac{x-μ}{\frac{σ}{\sqrt{n} } }[/tex]^{2}[/tex]

To study the population mean, μ, of a normal distribution with known variance, [tex]σ^{2}[/tex], the most accurate test to apply is a Z-test with a standard normal distribution.

Next is the significance level

α: 0,06

To establish the type of critical region/ rejection region we have to take a look at the null hypothesis because this region is determined under the assumption that the null hypothesis is true. In this case, since the hypothesis is = vs ≠, our critical region will be two-tailed. When we use a two-tailed test, the level of significance split between the two tails. So instead of using 6% whole, we will use 0,03 for each tail.

Remember, this distribution is simetric around cero so the left and right values are going to be -/+ Zc (see graphic)

For the left tail the critical value will be:

Zc1: [tex]Z_{α/2} = Z_{0.06/2} = Z_{0.03}[/tex] = - 1,88

For the right tail the critical value will be:

CV2: [tex]Z_{1-α/2} = Z_{1-0,03} = Z_{0,97}[/tex] = 1,88

I've attached an example using the Z-table with the right entry of the table since the distribution is symmetric you just need to look for one of the values and then change its sign for the other critical value. Just look for the probability in the center of the table and then look at the margins for the correspondent Zc

As for the values of Z, both 1,88 and 1,89 are close to the probability of 0,97. I've chosen 1,88 since is the closest.

I hope you have a SUPER day!