Question

Researchers at the University of Washington and Harvard University analyzed records of breast cancer screening and diagnostic evaluations.† Discussing the benefits and downsides of the screening process, the article states that although the rate of false-positives is higher than previously thought, if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall, but the rate of missed cancers would rise.

Suppose that such a screening test 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.)

(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?

Type I

or

Type II    

(b) Describe a Type I error in the context of this problem, and discuss the consequences of making a Type I error.

- Doctors may come to the conclusion that cancer is present when, in fact, it is not. Treatment may be started when, in fact, it is not necessary.

- Doctors may come to the conclusion that cancer is present when, in fact, it is not. No treatment will be prescribed when, in fact, treatment is necessary.

- Doctors may come to the conclusion that no cancer is present when, in fact, it is present. No treatment will be prescribed when, in fact, treatment is necessary.

- Doctors may come to the conclusion that no cancer is present when, in fact, it is present. Treatment may be started when, in fact, it is not necessary.

(c)

Describe a Type II error in the context of this problem, and discuss the possible consequences of making a Type II error.

- Doctors may come to the conclusion that cancer is present when, in fact, it is not. Treatment may be started when, in fact, it is not necessary.

- Doctors may come to the conclusion that cancer is present when, in fact, it is not. No treatment will be prescribed when, in fact, treatment is necessary.

- Doctors may come to the conclusion that no cancer is present when, in fact, it is present. No treatment will be prescribed when, in fact, treatment is necessary.

- Doctors may come to the conclusion that no cancer is present when, in fact, it is present. Treatment may be started when, in fact, it is not necessary.

Answer 1

(a) Type I, because we are rejecting the null hypothesis (H₀) when, in fact, H₀ is true.
(b) Doctors may come to the conclusion that cancer is present when, in fact, it is not. Treatment may be started when, in fact, it is not necessary, since by definition, we reject the null hypothesis (H₀) when, in fact, H₀ is true.
(c) Doctors may come to the conclusion that no cancer is present when, in fact, it is present. No treatment will be prescribed when, in fact, treatment is necessary, since by definition, we fail to reject the null hypothesis (H₀) when, in fact, H₀ is not true.