Question 1170282: In the discussion of Nissan's Leaf, we looked at two hypothesis tests for the same 6 measurements of the range. The first treated Nissan's claim as something to prove and found the claim was not proved with this data. The second test gave Nissan the benefit of the doubt and found we would not reject their claim based on this data. The conclusions appear contradictory. This chart illustrates assuming the population mean is 74 or 77 miles. null and competing hypotheses
As a statistician, explain this to your client using a court trial as a parallel.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely. Let's explain this apparent contradiction using a court trial analogy, making it clear to your client how hypothesis testing works and why the conclusions differ.
**The Court Trial Analogy**
Imagine a court trial where we're trying to determine if Nissan's claim about the Leaf's range is "true" or "false" based on our sample data.
**Scenario 1: We're Trying to Prove Nissan's Claim (Prosecution)**
* **Null Hypothesis (H0):** Nissan's claim is false (e.g., the average range is less than 77 miles).
* **Alternative Hypothesis (H1):** Nissan's claim is true (e.g., the average range is 77 miles or more).
* **Our Role:** We're the prosecution, trying to prove that Nissan's claim is *false*.
* **Burden of Proof:** We have to provide strong evidence to convince the "jury" (our statistical test) that Nissan's claim is wrong.
* **Outcome:** If we don't have enough evidence to confidently reject the null hypothesis (that Nissan's claim is false), we say "we fail to reject the null hypothesis." In court terms, it's like saying "not guilty," meaning we didn't prove their claim was wrong, but it doesn't mean their claim is automatically true.
**Scenario 2: We're Giving Nissan the Benefit of the Doubt (Defense)**
* **Null Hypothesis (H0):** Nissan's claim is true (e.g., the average range is 77 miles).
* **Alternative Hypothesis (H1):** Nissan's claim is false (e.g., the average range is less than 77 miles).
* **Our Role:** We're the defense, giving Nissan the benefit of the doubt.
* **Burden of Proof:** The "prosecution" (the data) has to provide strong evidence to convince the "jury" (our statistical test) that Nissan's claim is wrong.
* **Outcome:** If the "prosecution" doesn't have enough evidence to confidently reject the null hypothesis (that Nissan's claim is true), we say "we fail to reject the null hypothesis." In court terms, it's like saying "not guilty," meaning the data didn't prove their claim was false, but it doesn't mean their claim is automatically true.
**Why the "Contradiction" Isn't Real**
* The apparent contradiction stems from the difference in the burden of proof and the wording of the conclusions.
* In the first scenario, we couldn't *prove* Nissan's claim. In the second, we couldn't *disprove* it.
* "Failing to reject" doesn't mean "accepting." It means we didn't find enough evidence to say the opposite.
* It's like saying, "We didn't find them guilty," versus "We found them innocent." They are not the same.
* The placement of the claim in the null hypothesis, changes what we are attempting to prove.
* The data is the same, but the question being asked is different.
**Key Takeaways for Your Client**
* Hypothesis testing is about assessing evidence against a specific claim (the null hypothesis).
* The conclusion depends on the direction of the test and what we're trying to prove.
* "Failing to reject" doesn't validate a claim; it simply means we lack sufficient evidence to refute it.
* The null hypothesis is the statement that is assumed to be true until evidence indicates otherwise.
By using this court trial analogy, your client should understand that the two tests aren't contradictory. They simply reflect different perspectives and burdens of proof.
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