The College Board states that the average math SAT score is 514 with a standard deviation of 117 . Colleen gathered data from 50 students in her graduating class and found the average score to be 523 . She thinks that her class's math SAT score is different from the average. Which of the following are the correct null hypothesis and alternate hypothesis? \( \mathrm{H}_{0}: \mu=514 ; \mathrm{H}_{\mathrm{a}}: \mu>514 \) \( \mathrm{H}_{0}: \mu=514 ; \mathrm{H}_{\mathrm{a}}: \mu=523 \) \( \mathrm{H}_{0}: \mu=514 ; \mathrm{H}_{\mathrm{a}}: \mu=514 \) \( \mathrm{H}_{0}: \mu=514 ; \mathrm{H}_{\mathrm{a}}: \mu=523 \)

To determine the correct null hypothesis (\(H_0\)) and alternate hypothesis (\(H_a\)) for Colleen's situation, we need to understand the context of her claim. 1. **Null Hypothesis (\(H_0\))**: This is a statement that there is no effect or no difference. In this case, it would state that the average math SAT score of Colleen's class is equal to the national average, which is 514. 2. **Alternate Hypothesis (\(H_a\))**: This is a statement that indicates the presence of an effect or a difference. Colleen believes that her class's average score is different from the national average. This means that the alternate hypothesis should reflect that the average score is not equal to 514. Given this understanding, the correct hypotheses are: - \(H_0: \mu = 514\) (the average score of Colleen's class is equal to the national average) - \(H_a: \mu \neq 514\) (the average score of Colleen's class is different from the national average) However, since "not equal to" is not one of the options provided, we can analyze the options given: 1. \(H_0: \mu = 514; H_a: \mu > 514\) - This suggests that the average is greater than 514, which is not what Colleen is testing for since she is looking for any difference. 2. \(H_0: \mu = 514; H_a: \mu = 523\) - This is incorrect because the alternate hypothesis should not specify a single value. 3. \(H_0: \mu = 514; H_a: \mu = 514\) - This is incorrect as both hypotheses are the same. 4. \(H_0: \mu = 514; H_a: \mu = 523\) - This is incorrect because the alternate hypothesis should indicate a difference, not a specific value. Since none of the options correctly represent the hypothesis test for a two-tailed test (which is what Colleen's claim suggests), the closest option that reflects a difference (though not perfectly) is the first one, but it is still not entirely correct. In conclusion, the correct hypotheses should be: - \(H_0: \mu = 514\) - \(H_a: \mu \neq 514\) However, based on the provided options, none are entirely correct.