Answer :
To solve the problem and determine the null and alternative hypotheses, we need to understand the specific claim being tested. Here's a step-by-step explanation:
1. Identify the Claim: The claim states that the average amount spent on food per week by a family is [tex]$195. The home economist suspects that the true average is actually lower than $[/tex]195.
2. Define the Null Hypothesis ([tex]\(H_0\)[/tex]): The null hypothesis is a statement that there's no effect or no difference, and it's what you're testing against. It represents the status quo. For this problem, the null hypothesis would be that the true average amount spent is indeed [tex]$195. Therefore, \(H_0: \mu = 195\).
3. Define the Alternative Hypothesis (\(H_1\)): The alternative hypothesis is what you suspect might actually be true. In this scenario, the home economist suspects that the average is lower than $[/tex]195. This gives the alternative hypothesis: [tex]\(H_1: \mu < 195\)[/tex].
4. Summary of Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 195\)[/tex] (the average spending is [tex]$195)
- Alternative Hypothesis (\(H_1\)): \(\mu < 195\) (the average spending is less than $[/tex]195)
Thus, the correct choice for the hypotheses is:
- [tex]\(H_0: \mu = 195, H_1: \mu < 195\)[/tex]
This setup allows you to conduct a statistical test to determine if there is enough evidence to support the home economist's claim that the true average spending is indeed less than $195 when considering the given level of significance (0.01).
1. Identify the Claim: The claim states that the average amount spent on food per week by a family is [tex]$195. The home economist suspects that the true average is actually lower than $[/tex]195.
2. Define the Null Hypothesis ([tex]\(H_0\)[/tex]): The null hypothesis is a statement that there's no effect or no difference, and it's what you're testing against. It represents the status quo. For this problem, the null hypothesis would be that the true average amount spent is indeed [tex]$195. Therefore, \(H_0: \mu = 195\).
3. Define the Alternative Hypothesis (\(H_1\)): The alternative hypothesis is what you suspect might actually be true. In this scenario, the home economist suspects that the average is lower than $[/tex]195. This gives the alternative hypothesis: [tex]\(H_1: \mu < 195\)[/tex].
4. Summary of Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 195\)[/tex] (the average spending is [tex]$195)
- Alternative Hypothesis (\(H_1\)): \(\mu < 195\) (the average spending is less than $[/tex]195)
Thus, the correct choice for the hypotheses is:
- [tex]\(H_0: \mu = 195, H_1: \mu < 195\)[/tex]
This setup allows you to conduct a statistical test to determine if there is enough evidence to support the home economist's claim that the true average spending is indeed less than $195 when considering the given level of significance (0.01).
