Describe a Type I & Type II Error for each scenario.
$\left(C\right)$ A new cancer screening test reports whether a patient has cancer.

• A. Type I: The test is positive but the patient doesn't have cancer. Type II: The test is negative but the patient has cancer.
• B. Type I: The test is negative but the patient has cancer. Type II: The test is positive but the patient doesn't have cancer.
• C. Type I: The test is positive and the patient has cancer. Type II: The test is negative and the patient doesn't have cancer.
• D. Type I: The test is negative and the patient doesn't have cancer. Type II: The test is positive and the patient has cancer.

Answer: (A)

Verified step by step guidance

1. Step 1: Identify the null hypothesis (H0) and the alternative hypothesis (H1) based on the problem context. Here, the hypotheses relate to the parameter P, where H0: P = 0.85 and H1: P ≠ 0.85 (or specifically P > 0.85 or P < 0.85 depending on the scenario).
2. Step 2: Understand what a Type I error means: it occurs when we reject the null hypothesis H0 when it is actually true. In this context, a Type I error would be concluding that P is different from 0.85 (e.g., P > 0.85 or P < 0.85) when in fact P = 0.85.
3. Step 3: Understand what a Type II error means: it occurs when we fail to reject the null hypothesis H0 when the alternative hypothesis H1 is true. Here, a Type II error would be concluding that P = 0.85 when in fact P is greater than or less than 0.85.
4. Step 4: For each scenario, match the statements to the errors: for example, 'We conclude that P < 0.85 when actually P = 0.85' is a Type I error because we rejected H0 incorrectly; 'We conclude that P = 0.85 when actually P < 0.85' is a Type II error because we failed to reject H0 when it was false.
5. Step 5: Finally, consider the seriousness of the errors as given. If Type I is more serious, focus on minimizing false positives (rejecting H0 incorrectly). If Type II is more serious, focus on minimizing false negatives (failing to reject H0 when it is false). This helps interpret which error corresponds to which conclusion in the problem.