Answer :
In hypothesis testing, we are trying to determine if there is enough statistical evidence to support a certain belief or hypothesis about a population parameter. In this case, the research team is investigating whether a modification in the manufacturing process for fiber optic cables has increased the mean transmission distance.
Step-by-Step Explanation:
Define the Null and Alternative Hypotheses:
The null hypothesis (H0) represents the status quo or the initial claim that there is no change or effect. For this problem, the null hypothesis states that the mean transmission distance, [tex]\mu[/tex], is equal to 58 km, which is the original mean distance.
The alternative hypothesis (Ha) represents what we want to test, in this case, that the modification has resulted in a longer mean transmission distance. Thus, the alternative hypothesis states that the mean transmission distance [tex]\mu[/tex] is greater than 58 km.
Using mathematical notation, the hypotheses are:
[tex]H0: \mu = 58[/tex]
[tex]Ha: \mu > 58[/tex]Choosing the Correct Option:
From the given choices, the correct option that matches the defined hypotheses is:
H0: [tex]\mu = 58[/tex] versus Ha: [tex]\mu > 58[/tex]
Context and Details:
Who: A research team trying to improve fiber optic cable transmission.
What: Testing if a modification in cable manufacturing increases the mean transmission.
When/Where: Not specified, but typically done following batches created with the new process.
Why: To ensure that the new manufacturing process provides a significant improvement in the transmission distance which can be beneficial for usage in communications technology.
How: By comparing the sample mean transmission distance of 59.1 km to the historical mean of 58 km using statistical tests.
The hypotheses are fundamental in setting up a statistical test to evaluate the effectiveness of the new process and guide decision-making based on data collected from fiber optic cables.
