To test the claim that the proportion of Population 1 is no less than the proportion of Population 2 at a 10% level of significance, we first need to set up our hypotheses.
### Step 1: Define the Hypotheses
Null Hypothesis ([tex]\(H_0\)[/tex]): The proportion of Population 1 [tex]\((p_1)\)[/tex] is greater than or equal to the proportion of Population 2 [tex]\((p_2)\)[/tex].
[tex]\[ H_0: p_1 \geq p_2 \][/tex]
Alternative Hypothesis ([tex]\(H_a\)[/tex]): The proportion of Population 1 [tex]\((p_1)\)[/tex] is less than the proportion of Population 2 [tex]\((p_2)\)[/tex].
[tex]\[ H_a: p_1 < p_2 \][/tex]
### Step 2: State the Sample Size Requirement
For the z-test for comparing two proportions, the sample size requirements can be assumed to be large enough because we usually consider the conditions:
- [tex]\( n_1 \times p_1 \hat{}\)[/tex] and [tex]\( n_1 \times (1 - p_1 \hat{}) \)[/tex] are both greater than 5
- [tex]\( n_2 \times p_2 \hat{}\)[/tex] and [tex]\( n_2 \times (1 - p_2 \hat{}) \)[/tex] are both greater than 5
Since these conditions are given in the problem and provided solutions show that the sample sizes are sufficient, we proceed with the hypothesis test based on this data.
So, to summarize:
[tex]\[ H_0: p_1 \geq p_2 \][/tex]
[tex]\[ H_a: p_1 < p_2 \][/tex]
These form the basis for our hypothesis test.
