A public health agency claimed last year that the proportion of children vaccinated for measles met their goal of 85%, however, they want to test if the current proportion falls shy of that goal. What are the Type I & Type II Errors? Which is more serious?

• A. We conclude that P > 0.85 when actually P = 0.85We conclude that P = 0.85 when actually P > 0.85Type I is more serious
• B. We conclude that P < 0.85 when actually P = 0.85We conclude that P = 0.85 when actually P < 0.85Type I is more serious
• C. We conclude that P > 0.85 when actually P = 0.85We conclude that P = 0.85 when actually P > 0.85Type II is more serious
• D. We conclude that P < 0.85 when actually P = 0.85We conclude that P = 0.85 when actually P < 0.85Type II is more serious

Answer: (D)

Verified step by step guidance

1. Step 1: Define the null hypothesis (H_0) and the alternative hypothesis (H_a). Since the agency claimed the proportion of vaccinated children is 85%, the null hypothesis is \(H_0: P = 0.85\). The alternative hypothesis, testing if the current proportion falls short, is \(H_a: P < 0.85\).
2. Step 2: Understand Type I error: This occurs when we reject the null hypothesis even though it is true. In this context, a Type I error means concluding that the proportion of vaccinated children is less than 0.85 when in fact it is exactly 0.85.
3. Step 3: Understand Type II error: This happens when we fail to reject the null hypothesis even though the alternative is true. Here, a Type II error means concluding that the proportion is 0.85 when actually it is less than 0.85.
4. Step 4: Identify the errors in terms of the problem statements: Type I error corresponds to concluding \(P < 0.85\) when actually \(P = 0.85\). Type II error corresponds to concluding \(P = 0.85\) when actually \(P < 0.85\).
5. Step 5: Consider which error is more serious: Since the agency wants to ensure the vaccination rate does not fall below 85%, failing to detect a drop (Type II error) could have more serious public health consequences. Therefore, Type II error is more serious in this context.