Question 1163235: The screening process for detecting a rare disease is not perfect. Researchers have developed a blood test that is considered fairly reliable. It gives a positive reaction in 98.2% of the people who have that disease. However, it erroneously gives a positive reaction in 1.3% of the people who do not have the disease.
a. Calculate the probability of a Type I error and round your answer to 3 decimal places.
b. Calculate the probability of a Type II error and round your answer to 3 decimal places.
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website! In hypothesis testing, a Type I error and a Type II error represent the two ways a test can give an incorrect result. To solve this, we first define our "hypotheses" in the context of the disease:
* **$H_0$ (Null Hypothesis):** The person does **not** have the disease.
* **$H_A$ (Alternative Hypothesis):** The person **has** the disease.
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### a. Calculate the probability of a Type I error
A **Type I error** (often denoted as $\alpha$) occurs when we reject the null hypothesis even though it is true. In medical terms, this is a **False Positive**.
* This happens when the test says a person **has** the disease, but they actually **do not**.
* The problem states the test erroneously gives a positive reaction in **1.3%** of people who do not have the disease.
**Calculation:**
$$P(\text{Type I Error}) = 1.3\% = 0.013$$
> **Probability of a Type I error: 0.013**
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### b. Calculate the probability of a Type II error
A **Type II error** (often denoted as $\beta$) occurs when we fail to reject the null hypothesis even though it is false. In medical terms, this is a **False Negative**.
* This happens when the test says a person **does not** have the disease, but they actually **do**.
* The problem states the test correctly identifies **98.2%** of people who have the disease (this is the "Power" or Sensitivity of the test).
* The Type II error is the remaining percentage of people with the disease who were missed by the test.
**Calculation:**
$$P(\text{Type II Error}) = 100\% - 98.2\%$$
$$P(\text{Type II Error}) = 1.8\% = 0.018$$
> **Probability of a Type II error: 0.018**
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