Answer :
To solve this question, we need to determine the correct null hypothesis (H₀) and alternative hypothesis (Hₐ) based on Alyssa's belief and given data.
### Step-by-step solution:
1. Understanding the Hypotheses:
- The null hypothesis (H₀) is a statement that there is no effect or difference. It usually represents the status quo or a baseline value.
- The alternative hypothesis (Hₐ) is what you want to prove. It represents a new effect or difference that you suspect.
2. Given Information:
- The mean price of a prom dress from a large survey is [tex]$195.00.
- Alyssa collected data at her school and found an average price of $[/tex]208.00.
- Alyssa thinks that students at her school spent more than [tex]$195.00 on prom dresses.
3. Formulating the Hypotheses:
- The null hypothesis (H₀) should state that the mean price of a prom dress at Alyssa's school is equal to the general mean price, $[/tex]195.00.
[tex]\[
H₀: \mu = 195
\][/tex]
- The alternative hypothesis (Hₐ) should state that the mean price of a prom dress at Alyssa's school is greater than the general mean price, $195.00.
[tex]\[
Hₐ: \mu > 195
\][/tex]
4. Matching the Hypotheses to the Given Choices:
- Option 1: [tex]\( H_0: \not=195 ; H_a: \not>195 \)[/tex]
- Option 2: [tex]\( H_{0}: \mu \neq 195 ; H_{a}: \mu=208 \)[/tex]
- Option 3: [tex]\( H_{0}: \mu=195 ; H_{a}: \mu \neq 195 \)[/tex]
- Option 4: [tex]\( H_{0}: \mu<195 ; H_{a}: \mu \geq 208 \)[/tex]
Considering the correct formulation of hypotheses, we match the pairs:
- None of the given options perfectly match the appropriate alternative hypothesis [tex]\(H_{a}: \mu > 195\)[/tex].
If there were additional choices or a misinterpretation of the mentioned labels (like assuming labels `H_ₐ` instead of `H₃`), be sure to select the closest matching option. For proper scientific interpretation, check for accurate hypothesis formulation.
Given the choices, option 3 [tex]\( H_{0}: \mu = 195; H_{a}: \mu \neq 195\)[/tex] aligns closest yet doesn't convey the exact "greater than" condition accurately sought. For the correct depiction:
- You could recalibrate or search more innovative approaches to align conducive selections in real scenarios.
### Correct Hypotheses:
[tex]\[
H_0: \mu = 195
\][/tex]
[tex]\[
H_a: \mu > 195
\][/tex]
Final Insight:
For exact framework-based academic setup, attentively confirming logical draft peeking proposed models and standard mathematical vicinity crucially aids decision-making.
### Step-by-step solution:
1. Understanding the Hypotheses:
- The null hypothesis (H₀) is a statement that there is no effect or difference. It usually represents the status quo or a baseline value.
- The alternative hypothesis (Hₐ) is what you want to prove. It represents a new effect or difference that you suspect.
2. Given Information:
- The mean price of a prom dress from a large survey is [tex]$195.00.
- Alyssa collected data at her school and found an average price of $[/tex]208.00.
- Alyssa thinks that students at her school spent more than [tex]$195.00 on prom dresses.
3. Formulating the Hypotheses:
- The null hypothesis (H₀) should state that the mean price of a prom dress at Alyssa's school is equal to the general mean price, $[/tex]195.00.
[tex]\[
H₀: \mu = 195
\][/tex]
- The alternative hypothesis (Hₐ) should state that the mean price of a prom dress at Alyssa's school is greater than the general mean price, $195.00.
[tex]\[
Hₐ: \mu > 195
\][/tex]
4. Matching the Hypotheses to the Given Choices:
- Option 1: [tex]\( H_0: \not=195 ; H_a: \not>195 \)[/tex]
- Option 2: [tex]\( H_{0}: \mu \neq 195 ; H_{a}: \mu=208 \)[/tex]
- Option 3: [tex]\( H_{0}: \mu=195 ; H_{a}: \mu \neq 195 \)[/tex]
- Option 4: [tex]\( H_{0}: \mu<195 ; H_{a}: \mu \geq 208 \)[/tex]
Considering the correct formulation of hypotheses, we match the pairs:
- None of the given options perfectly match the appropriate alternative hypothesis [tex]\(H_{a}: \mu > 195\)[/tex].
If there were additional choices or a misinterpretation of the mentioned labels (like assuming labels `H_ₐ` instead of `H₃`), be sure to select the closest matching option. For proper scientific interpretation, check for accurate hypothesis formulation.
Given the choices, option 3 [tex]\( H_{0}: \mu = 195; H_{a}: \mu \neq 195\)[/tex] aligns closest yet doesn't convey the exact "greater than" condition accurately sought. For the correct depiction:
- You could recalibrate or search more innovative approaches to align conducive selections in real scenarios.
### Correct Hypotheses:
[tex]\[
H_0: \mu = 195
\][/tex]
[tex]\[
H_a: \mu > 195
\][/tex]
Final Insight:
For exact framework-based academic setup, attentively confirming logical draft peeking proposed models and standard mathematical vicinity crucially aids decision-making.
