Answer :
To determine the correct setup for the null and alternative hypotheses, we need to consider the claim being tested and the typical structure of hypothesis testing.
### Step-by-step Solution:
1. Identify the Claim:
The real estate agent claims that the average price of a home in a certain zip code is less than [tex]$250,000. This claim forms the basis of our alternative hypothesis.
2. Define the Hypotheses:
- Null Hypothesis (\(H_0\)): This is a statement of no effect or no difference, and it often assumes equality or a situation that we want to test against. In this case, the null hypothesis is that the average home price is greater than or equal to $[/tex]250,000.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): This represents the claim that we want to test. Here, it suggests that the average home price is less than [tex]$250,000.
3. Formulate the Hypotheses:
- \(H_0: \mu \geq 250,000\)
- \(H_a: \mu < 250,000\)
4. Determine the Type of Test:
The alternative hypothesis involves the phrase "less than," which indicates a left-tailed test. Left-tailed tests are used when the alternative hypothesis is testing if a parameter is less than a specified value.
5. Conclusion:
Based on these definitions, the correct setup for the null and alternative hypotheses for this test is:
\[
H_0: \mu \geq 250,000 ; H_a: \mu < 250,000, \quad \text{which is a left-tailed test.}
\]
Therefore, the appropriate setup for testing the real estate agent's claim is a left-tailed test, where the null hypothesis states the average price is at least $[/tex]250,000, and the alternative hypothesis states it is less than $250,000.
### Step-by-step Solution:
1. Identify the Claim:
The real estate agent claims that the average price of a home in a certain zip code is less than [tex]$250,000. This claim forms the basis of our alternative hypothesis.
2. Define the Hypotheses:
- Null Hypothesis (\(H_0\)): This is a statement of no effect or no difference, and it often assumes equality or a situation that we want to test against. In this case, the null hypothesis is that the average home price is greater than or equal to $[/tex]250,000.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): This represents the claim that we want to test. Here, it suggests that the average home price is less than [tex]$250,000.
3. Formulate the Hypotheses:
- \(H_0: \mu \geq 250,000\)
- \(H_a: \mu < 250,000\)
4. Determine the Type of Test:
The alternative hypothesis involves the phrase "less than," which indicates a left-tailed test. Left-tailed tests are used when the alternative hypothesis is testing if a parameter is less than a specified value.
5. Conclusion:
Based on these definitions, the correct setup for the null and alternative hypotheses for this test is:
\[
H_0: \mu \geq 250,000 ; H_a: \mu < 250,000, \quad \text{which is a left-tailed test.}
\]
Therefore, the appropriate setup for testing the real estate agent's claim is a left-tailed test, where the null hypothesis states the average price is at least $[/tex]250,000, and the alternative hypothesis states it is less than $250,000.
