To determine whether Colleen's class's math SAT scores differ from the average provided by the College Board, we need to set up the appropriate null and alternate hypotheses:
1. Null Hypothesis ([tex]\( H_0 \)[/tex]): This is the hypothesis that there is no difference in the population parameter. It typically represents the status quo or a position of no effect. In Colleen's case, it would be stating that the average math SAT score of her class is not different from the average math SAT score given by the College Board, which is 514.
2. Alternate Hypothesis ([tex]\( H_a \)[/tex]): This represents what Colleen is trying to prove - that her class's average math SAT score is different from the given average. This hypothesis indicates that there is a difference.
Given Colleen's belief that her class's math SAT score is different from the average, we need a two-tailed test. Thus, the hypotheses can be defined as follows:
- The null hypothesis ( [tex]\( H_0 \)[/tex] ): [tex]\( \mu = 514 \)[/tex], which means the average math SAT score is 514.
- The alternate hypothesis ( [tex]\( H_a \)[/tex] ): [tex]\( \mu \neq 514 \)[/tex], which means the average math SAT score is not 514.
From the provided options, the correct hypotheses are:
- [tex]\( H_0: \mu = 514 \)[/tex]
- [tex]\( H_a: \mu \neq 514 \)[/tex]
Thus, the correct option from the given choices is:
[tex]\[ H_0: \mu = 514; H_a: \mu \neq 514 \][/tex]
This choice aligns with what is needed to test whether there is a statistically significant difference in the average scores.
Note that the number '3' supplied in the result is unrelated to this specific hypothesis decision and might relate to an index in a list of correct options (such as the third option being correct), which aligns with the chosen hypothesis setup.
