To determine the correct null and alternate hypotheses for the given scenario, we need to follow a systematic approach.
1. Understanding the Claim:
Jonathan believes that his football team buddies watch less television than the average American.
2. Identifying the Population Mean:
The average time Americans watch television each weekday is given as 2.7 hours, with a standard deviation of 0.2 hours.
3. Sample Data:
Jonathan gathered data from 40 football teammates and calculated their mean television watching time to be 2.3 hours.
4. Formulating Hypotheses:
- The null hypothesis ( [tex]\(H_0\)[/tex] ) is a statement that there is no effect or no difference, and it is generally presumed to be true until statistical evidence indicates otherwise. In this context, the null hypothesis should reflect that the football team buddies do not watch less television than the average American, meaning their average [tex]\( \mu \)[/tex] is equal to 2.7 hours.
- The alternate hypothesis ( [tex]\(H_a\)[/tex] ) is what we want to test for; it represents Jonathan's belief. Therefore, it should state that the football team buddies watch less television than the average American, meaning their average [tex]\( \mu \)[/tex] is less than 2.7 hours.
Based on these points:
- The null hypothesis [tex]\( H_0 \)[/tex]: [tex]\( \mu = 2.7 \)[/tex]
- The alternate hypothesis [tex]\( H_a \)[/tex]: [tex]\( \mu < 2.7 \)[/tex]
Thus, the correct pair of hypotheses is:
[tex]\[ H_0: \mu = 2.7 \][/tex]
[tex]\[ H_a: \mu < 2.7 \][/tex]
From the given options, the correct answer is:
[tex]\[ H_0: \mu = 2.7 ; H_a: \mu < 2.7 \][/tex]
So, among the listed options, it should be:
[tex]\[ H_0: \mu = 2.7 ; H_a: \mu < 2.7 \][/tex]
