Answer :
To address Colleen's hypothesis regarding her class's math SAT scores, we need to first establish the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)) correctly. Here's how you can approach this:
1. Null Hypothesis (\(H_0\)): This hypothesis represents the status quo or a statement of no effect or no difference. It typically posits that any kind of difference or effect is due to random chance. In this context, the null hypothesis would state that the mean SAT score of Colleen's class is the same as the national average reported by the College Board, which is 514.
So, \(H_0: \mu = 514\).
2. Alternative Hypothesis (\(H_a\)): This hypothesis represents the claim or the significant effect that Colleen is trying to establish. Colleen thinks her class's SAT score is different from the average; she doesn’t specify whether she thinks it’s specifically higher or lower, just different. Thus, her hypothesis is non-directional.
So, \(H_a: \mu \neq 514\).
Now, reviewing the given options:
- \(H_0: \mu = 514 ; H_a: \mu > 514\)
This option suggests a one-tailed test where the alternative hypothesis is that the mean score is greater than the national average. This does not align with Colleen’s belief of the score being different without specifying a direction.
- \(H_0: \mu = 514 ; H_a: \mu = 523\)
This option incorrectly formulates the alternative hypothesis as a specific value, which isn't standard practice for hypothesis testing.
- \(H_0: \mu = 514 ; H_a: \mu = 514\)
This option incorrectly states the alternative hypothesis to be the same as the null hypothesis, which makes no logical sense as it doesn’t actually propose an alternative.
- \(H_0: \mu = 514 ; H_a: \mu \neq 514\)
Although this exact option is not listed, it's the correct way to state the hypotheses logically based on Colleen’s belief that her class's average SAT score may be different from the national average.
Given the correct forms of hypotheses in a statistical context and consultation with the given options, none of the provided options precisely match the correct representation. The correct hypotheses should be:
- [tex]\(H_0: \mu = 514 ; H_a: \mu \neq 514\)[/tex].
1. Null Hypothesis (\(H_0\)): This hypothesis represents the status quo or a statement of no effect or no difference. It typically posits that any kind of difference or effect is due to random chance. In this context, the null hypothesis would state that the mean SAT score of Colleen's class is the same as the national average reported by the College Board, which is 514.
So, \(H_0: \mu = 514\).
2. Alternative Hypothesis (\(H_a\)): This hypothesis represents the claim or the significant effect that Colleen is trying to establish. Colleen thinks her class's SAT score is different from the average; she doesn’t specify whether she thinks it’s specifically higher or lower, just different. Thus, her hypothesis is non-directional.
So, \(H_a: \mu \neq 514\).
Now, reviewing the given options:
- \(H_0: \mu = 514 ; H_a: \mu > 514\)
This option suggests a one-tailed test where the alternative hypothesis is that the mean score is greater than the national average. This does not align with Colleen’s belief of the score being different without specifying a direction.
- \(H_0: \mu = 514 ; H_a: \mu = 523\)
This option incorrectly formulates the alternative hypothesis as a specific value, which isn't standard practice for hypothesis testing.
- \(H_0: \mu = 514 ; H_a: \mu = 514\)
This option incorrectly states the alternative hypothesis to be the same as the null hypothesis, which makes no logical sense as it doesn’t actually propose an alternative.
- \(H_0: \mu = 514 ; H_a: \mu \neq 514\)
Although this exact option is not listed, it's the correct way to state the hypotheses logically based on Colleen’s belief that her class's average SAT score may be different from the national average.
Given the correct forms of hypotheses in a statistical context and consultation with the given options, none of the provided options precisely match the correct representation. The correct hypotheses should be:
- [tex]\(H_0: \mu = 514 ; H_a: \mu \neq 514\)[/tex].
