Answer :
To solve this problem, we need to set up the correct null and alternate hypotheses based on the scenario provided. Alyssa believes students at her school spend more on prom dresses than the mean price found in the survey. Let's go through the steps to determine the appropriate hypotheses:
1. Understand the Scenario:
- A national survey found the mean price of a prom dress to be [tex]$195.00.
- Alyssa believes students at her school spend more than this amount.
- She collected data from 20 students at her school, with an average dress price of $[/tex]208.00.
2. Define the Null Hypothesis (H₀):
- The null hypothesis represents a statement of no effect or no difference. It is what we assume to be true before collecting any evidence.
- In this case, the null hypothesis would be that there is no difference in the average dress price, i.e., students at Alyssa's school spend the same as the national average.
- Mathematically, the null hypothesis is: [tex]\( H_0: \mu = 195 \)[/tex]
3. Define the Alternative Hypothesis (Hₐ):
- The alternative hypothesis is what Alyssa is trying to prove. She believes students at her school spend more than the national average on prom dresses.
- Therefore, the alternative hypothesis would suggest that the mean price spent is greater than $195.00.
- Mathematically, the alternative hypothesis is: [tex]\( H_a: \mu > 195 \)[/tex]
4. Choose the Correct Option:
- We need to match our hypotheses to the options given:
- [tex]\( H_0: \mu = 195 ; H_a: \mu > 195 \)[/tex]
- [tex]\( H_0: \mu \neq 195 ; H_a: \mu = 208 \)[/tex]
- [tex]\( H_0: \mu = 195 ; H_a: \mu \neq 195 \)[/tex]
- [tex]\( H_0: \mu <195 ; H_a: \mu \geq 208 \)[/tex]
- The correct pair, based on our determined hypotheses, is:
- [tex]\( H_0: \mu = 195 ; H_a: \mu > 195 \)[/tex]
The correct null hypothesis and alternate hypothesis are:
[tex]\( H_0: \mu = 195 \)[/tex] and [tex]\( H_a: \mu > 195 \)[/tex].
1. Understand the Scenario:
- A national survey found the mean price of a prom dress to be [tex]$195.00.
- Alyssa believes students at her school spend more than this amount.
- She collected data from 20 students at her school, with an average dress price of $[/tex]208.00.
2. Define the Null Hypothesis (H₀):
- The null hypothesis represents a statement of no effect or no difference. It is what we assume to be true before collecting any evidence.
- In this case, the null hypothesis would be that there is no difference in the average dress price, i.e., students at Alyssa's school spend the same as the national average.
- Mathematically, the null hypothesis is: [tex]\( H_0: \mu = 195 \)[/tex]
3. Define the Alternative Hypothesis (Hₐ):
- The alternative hypothesis is what Alyssa is trying to prove. She believes students at her school spend more than the national average on prom dresses.
- Therefore, the alternative hypothesis would suggest that the mean price spent is greater than $195.00.
- Mathematically, the alternative hypothesis is: [tex]\( H_a: \mu > 195 \)[/tex]
4. Choose the Correct Option:
- We need to match our hypotheses to the options given:
- [tex]\( H_0: \mu = 195 ; H_a: \mu > 195 \)[/tex]
- [tex]\( H_0: \mu \neq 195 ; H_a: \mu = 208 \)[/tex]
- [tex]\( H_0: \mu = 195 ; H_a: \mu \neq 195 \)[/tex]
- [tex]\( H_0: \mu <195 ; H_a: \mu \geq 208 \)[/tex]
- The correct pair, based on our determined hypotheses, is:
- [tex]\( H_0: \mu = 195 ; H_a: \mu > 195 \)[/tex]
The correct null hypothesis and alternate hypothesis are:
[tex]\( H_0: \mu = 195 \)[/tex] and [tex]\( H_a: \mu > 195 \)[/tex].
