Math Problem Statement
A skeptical paranormal researcher claims that the proportion of Americans that have seen a UFO is less than 0.1% (p < 0.1%). An independent firm conducts a hypothesis test which supports his claim. If the true proportion of Americans who have seen a UFO is 0.15%, then which choice reflects what has occurred? Type I Error Type II Error Correct Decision
Solution
In this case, the hypothesis test is set up as follows:
- Null hypothesis (1% of Americans have seen a UFO).
- Alternative hypothesis (less than 1% of Americans have seen a UFO).
Now, according to the problem:
- The true proportion of Americans who have seen a UFO is (0.15%).
- The test supports the skeptical researcher's claim, meaning that the test concludes .
Analysis:
- A Type I error occurs when the null hypothesis is true, but the test incorrectly rejects it.
- A Type II error occurs when the null hypothesis is false, but the test fails to reject it.
- A correct decision occurs when the test's conclusion aligns with the true proportion.
Here, the true proportion is , which is slightly higher than 0.1%. Since the test incorrectly supports the claim that , it has failed to recognize that the true proportion is actually larger than the claimed 0.1%.
Thus, this situation reflects a Type I Error, because the test has incorrectly rejected a true null hypothesis.
Would you like more details or have any questions?
Here are five related questions to explore:
- What are the differences between Type I and Type II errors?
- How can the significance level () affect Type I errors?
- What steps can be taken to reduce Type II errors in hypothesis testing?
- Can a Type I error and Type II error occur in the same test?
- How is statistical power related to Type II errors?
Tip: Lowering the significance level (α) reduces the likelihood of a Type I error but increases the risk of a Type II error. Balancing these is key in hypothesis testing.
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Type I Error
Type II Error
Significance Level
Formulas
p-value comparison with alpha (significance level)
Theorems
Type I and Type II errors in statistical hypothesis testing
Suitable Grade Level
College/University level (Statistics)
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