Chapter 10.2, Problem 12E

Researchers at Boston’s Children’s Hospital and Harvard Medical School analyzed records of breast cancer screening and diagnostic evaluations (National Expenditure for False-Positive Mammograms and Breast Cancer Overdiagnoses Estimated at $4 Billion a Year,” Health Affairs [2015]: 576–583). Discussing the downsides of the screening process, the article states that the rate of false-positives is higher than previously thought, and that false-positives lead to unnecessary medical follow-up that can be costly.

Suppose that screening is used to decide between a null hypothesis of

H₀: no cancer is present

and an alternative hypothesis of

H_a: cancer is present.

(Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) (Hint: See Example 10.6.)

1. a. Would a false-positive (thinking that cancer is present when in fact it is not) be a Type I error or a Type II error?
2. b. Describe a Type I error in the context of this problem, and discuss the possible consequences of making a Type I error.
3. c. Describe a Type II error in the context of this problem, and discuss the possible consequences of making a Type II error.
4. d. Which type of error are the researchers concerned about when they say that false-positives lead to unnecessary medical follow-up? Explain why it would be reasonable to use a small significance level.