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?
[tex]$H _0: \mu=514 ; H _{ a }: \mu>514$[/tex]
[tex]$H _0: \mu=514 ; H _{ a }: \mu=523$[/tex]
[tex]$H _0: \mu=514 ; H _{ a }: \mu=514$[/tex]
[tex]$H _0: \mu=514 ; H _{ a }: \mu=523$[/tex]
Answer (2)
The null hypothesis states the population mean is 514: H 0 : μ = 514 .
The alternative hypothesis states the population mean is different from 514: H a : μ = 514 .
Therefore, the correct null and alternative hypotheses are H 0 : μ = 514 and H a : μ = 514 .
The correct answer is H 0 : μ = 514 ; H a : μ = 514 .
Explanation
Analyze the problem We need to determine 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 is 514. The alternative hypothesis represents Colleen's belief that her class's average score is different from the national average.
Define the null hypothesis The null hypothesis ( H 0 ) states that the population mean ( μ ) is equal to the stated average math SAT score, which is 514. Therefore, H 0 : μ = 514 .
Define the alternative hypothesis The alternative hypothesis ( H a ) states that the population mean ( μ ) is not equal to the stated average math SAT score, since Colleen thinks the score is different from the average. This is a two-tailed test. Therefore, H a : μ = 514 .
Compare with given options Comparing our hypotheses to the given options, none of them exactly match H 0 : μ = 514 and H a : μ = 514 . However, the closest representation, considering the context of hypothesis testing, would be to consider if the mean is simply different, without specifying if it's greater or equal to a specific value like 523. Therefore, we look for the option that has H 0 : μ = 514 and an alternative hypothesis that suggests a difference from 514.
State the correct hypotheses Since Colleen believes the average score is different from 514, the alternative hypothesis should reflect this. The correct hypotheses are: H 0 : μ = 514 H a : μ = 514
Examples
In hypothesis testing, we often want to determine if a sample's data provides enough evidence to reject a null hypothesis. For example, a company might claim their product lasts an average of 5 years. If we collect data showing the product lasts significantly longer or shorter, we can use hypothesis testing to see if the company's claim is likely false. This involves setting up null and alternative hypotheses and calculating a test statistic to make a decision.
The correct null hypothesis is H 0 : μ = 514 while the alternative hypothesis is H a : μ = 514 . This indicates that Colleen believes her class's average SAT score is different from the average score of 514. None of the provided options match this exactly, but the correct hypothesis conceptually supports her claim.
;
