Answer :
To solve this problem, we need to establish the correct null and alternate hypotheses based on the information provided:
1. Understanding the Context:
- We are told that the mean price of a prom dress across a large survey of 8,000 high school students is [tex]$195.00, with a standard deviation of $[/tex]12.00.
- Alyssa suspects that students at her school spend more on prom dresses, with her data from 20 students showing an average of [tex]$208.00.
2. Hypothesis Testing Basics:
- The null hypothesis (H₀) is a statement of no effect or no difference. In this context, it represents the status quo, which is the mean price of a prom dress being $[/tex]195.00 for high school students generally.
- The alternate hypothesis (Hₐ) is what you want to test; in this case, Alyssa's suspicion that students at her school spend more on prom dresses.
3. Setting up the Hypotheses:
- Null Hypothesis (H₀): The mean price of a prom dress is [tex]$195.00. This can be expressed as \( H₀: \mu = 195 \).
- Alternate Hypothesis (Hₐ): Alyssa believes that students at her school spend more than the average on prom dresses. This is expressed as \( Hₐ: \mu > 195 \).
4. Choosing the Hypotheses:
- Given Alyssa's belief that students spend more than the reported average, we want to test if the mean price is greater than $[/tex]195. Therefore, the correct pair of hypotheses is:
- [tex]\( H₀: \mu = 195 \)[/tex]
- [tex]\( Hₐ: \mu > 195 \)[/tex]
This matches the first option provided in the list:
- [tex]\( H _0: \mu=195 ; H _a: \mu>195 \)[/tex]
Hence, this is the correct null and alternate hypothesis for this scenario.
1. Understanding the Context:
- We are told that the mean price of a prom dress across a large survey of 8,000 high school students is [tex]$195.00, with a standard deviation of $[/tex]12.00.
- Alyssa suspects that students at her school spend more on prom dresses, with her data from 20 students showing an average of [tex]$208.00.
2. Hypothesis Testing Basics:
- The null hypothesis (H₀) is a statement of no effect or no difference. In this context, it represents the status quo, which is the mean price of a prom dress being $[/tex]195.00 for high school students generally.
- The alternate hypothesis (Hₐ) is what you want to test; in this case, Alyssa's suspicion that students at her school spend more on prom dresses.
3. Setting up the Hypotheses:
- Null Hypothesis (H₀): The mean price of a prom dress is [tex]$195.00. This can be expressed as \( H₀: \mu = 195 \).
- Alternate Hypothesis (Hₐ): Alyssa believes that students at her school spend more than the average on prom dresses. This is expressed as \( Hₐ: \mu > 195 \).
4. Choosing the Hypotheses:
- Given Alyssa's belief that students spend more than the reported average, we want to test if the mean price is greater than $[/tex]195. Therefore, the correct pair of hypotheses is:
- [tex]\( H₀: \mu = 195 \)[/tex]
- [tex]\( Hₐ: \mu > 195 \)[/tex]
This matches the first option provided in the list:
- [tex]\( H _0: \mu=195 ; H _a: \mu>195 \)[/tex]
Hence, this is the correct null and alternate hypothesis for this scenario.
