To find the critical [tex]\( z \)[/tex] value used to test the null hypothesis given the conditions [tex]\( \alpha = 0.08 \)[/tex] and the alternative hypothesis [tex]\( H_1: \mu \neq 3.24 \)[/tex], let's go through the solution step-by-step:
### Step 1: Understand the Problem
- We are conducting a hypothesis test where the null hypothesis is tested against the alternative hypothesis that the mean [tex]\( \mu \)[/tex] is not equal to 3.24.
- Since [tex]\( H_1: \mu \neq 3.24 \)[/tex] represents a two-tailed test, we will split the level of significance [tex]\( \alpha \)[/tex] into two tails.
### Step 2: Determine the Level of Significance Per Tail
- The total level of significance [tex]\( \alpha \)[/tex] is 0.08.
- For a two-tailed test, the significance level is evenly split between the two tails of the normal distribution.
[tex]\[
\frac{\alpha}{2} = \frac{0.08}{2} = 0.04
\][/tex]
### Step 3: Find the Critical [tex]\( z \)[/tex] Value(s)
- In a normal distribution, the critical [tex]\( z \)[/tex] value is determined using the cumulative distribution function (CDF). We need the [tex]\( z \)[/tex] value that corresponds to an area of [tex]\( 1 - 0.04 = 0.96 \)[/tex] from the left (for the upper tail) and [tex]\( 0.04 \)[/tex] from the left (for the lower tail).
- These critical [tex]\( z \)[/tex] values are the points beyond which the probability is [tex]\( \frac{\alpha}{2} \)[/tex] in each tail.
### Result
- The critical [tex]\( z \)[/tex] values corresponding to this distribution are approximately [tex]\( \pm 1.75 \)[/tex].
The correct answer from the given choices is [tex]\( \pm 1.75 \)[/tex].
