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 H0:p=0.001H_0: p = 0.001 (1% of Americans have seen a UFO).
  • Alternative hypothesis H1:p<0.001H_1: p < 0.001 (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 p=0.0015p = 0.0015 (0.15%).
  • The test supports the skeptical researcher's claim, meaning that the test concludes p<0.001p < 0.001.

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 p=0.0015p = 0.0015, which is slightly higher than 0.1%. Since the test incorrectly supports the claim that p<0.001p < 0.001, 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:

  1. What are the differences between Type I and Type II errors?
  2. How can the significance level (α\alpha) affect Type I errors?
  3. What steps can be taken to reduce Type II errors in hypothesis testing?
  4. Can a Type I error and Type II error occur in the same test?
  5. 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)