To perform a hypothesis test to determine if the machines are producing dresses with the correct mean length, we begin by setting our null and alternative hypotheses. Here is the detailed step-by-step solution:
Step 1: State the hypotheses
1. Null Hypothesis (H₀): This is the statement that there is no effect or no difference, and it represents the status quo. For this problem, we are testing whether the mean length of the dresses is equal to the designed length of 42 inches.
[tex]\[
H_0: \mu = 42
\][/tex]
2. Alternative Hypothesis (Hₐ): This is the statement that reflects what we are trying to find evidence for. In this case, we want to test if the mean length of the dresses is different from 42 inches, which could mean it is either longer or shorter. This is a two-tailed test because we are considering both possibilities.
[tex]\[
H_a: \mu \neq 42
\][/tex]
So, the complete hypotheses are:
[tex]\[
\begin{array}{l}
H_0: \mu = 42 \\
H_a: \mu \neq 42
\end{array}
\][/tex]
This forms the basis for our hypothesis test to determine whether the machines are producing dresses with the correct mean length.
