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?
A. [tex]$H _0: \mu=514 ; H _{ a }: \mu\ \textgreater \ 514$[/tex]
B. [tex]$H _0: \mu \neq 514 ; H _{ a }: \mu=523$[/tex]
C. [tex]$H _0: \mu=514 ; H _{ a }: \mu \neq 514$[/tex]
D. [tex]$H _0: \mu=514 ; H _{ a }: \mu=523$[/tex]
Answer (2)
The null hypothesis states that the population mean is equal to the national average: H 0 : μ = 514 .
The alternative hypothesis states that the population mean is not equal to the national average: H a : μ = 514 .
Therefore, the correct hypotheses are H 0 : μ = 514 ; H a : μ = 514 .
The correct answer is: H 0 : μ = 514 ; H a : μ = 514
Explanation
Problem Analysis We need to identify the correct null and alternative hypotheses for Colleen's claim. The null hypothesis represents the status quo, which in this case is that the average math SAT score for her class is the same as the national average. The alternative hypothesis represents Colleen's belief that her class's average is different from the national average.
Null Hypothesis The null hypothesis ( H 0 ) states that the population mean ( μ ) is equal to the national average, which is 514. Therefore, H 0 : μ = 514 .
Alternative Hypothesis The alternative hypothesis ( H a ) states that the population mean ( μ ) is not equal to the national average, since Colleen believes the average is different, and not specifically higher or lower. Therefore, H a : μ = 514 .
Conclusion Comparing our derived hypotheses with the given options, we find that the correct pair is: H 0 : μ = 514 ; H a : μ = 514
Examples
In hypothesis testing, we often want to determine if a sample mean is significantly different from a known population mean. For example, a researcher might want to test if a new teaching method improves test scores compared to the traditional method. The null hypothesis would be that there is no difference, and the alternative hypothesis would be that there is a difference. This type of hypothesis testing is widely used in various fields, including education, medicine, and business, to make informed decisions based on data.
Colleen's null hypothesis states that the average math SAT score is 514 ( H 0 : m u = 514 ), while the alternative hypothesis indicates that it is different ( H a : m u = 514 ). Therefore, the correct hypotheses are: H 0 : m u = 514 ; H a : m u = 514 . The correct answer is option C.
;
