Question

Type I and Type II Errors

The information for a particular area and radon testing is as follows:

• 3 of 100 homes have radon gas problems
• A test for radon gas is 85% accurate
• The test is performed in 10,000 homes in this area

a. Using the table below, create a matrix similar to the one in the Type I-Type II error notes.

1. Label the cells as true positives, false positives, false negatives, true negatives and Type I and Type II errors.
2. Complete the matrix to show the values for true positives, false positives (Type I), false negatives (Type II), and true negatives
3. Calculate the totals and add them to the matrix.  

b. What is the chance that a positive radon test really means that there is radon in that house? Show this as a percentage with 3 decimal places.

c. Assume that the test results were negative for a particular house.  What is the chance that there really is a radon problem in that house? Show this as a percentage with 3 decimal places.

Answer 1

	Test
			Positive	Negative
	Gas Problem	Yes	True Positive	False Negative (Type II error)
		No	False Positive (Type I error)	True Negative

total actual radon = 10000* 3/100 = 300

	Test
			Positive	Negative	Total
	Gas Problem	Yes	(85/100) * 300 = 255	300-255 = 45	(3/100) * 10000 = 300
		No	9700 - 8245 = 1455	(85/100) *9700 = 8245	10000 - 300 = 9700
		Total	1710	8290	10000

		Test
		Positive	Negative	Total
Gas Problem	Yes	255	45	300
	No	1455	8245	9700
	Total	1710	8290	10000

b)

chance that a positive radon test really means that there is radon in that house

= P(yes for Gas problem| test is positive)

= 255/1710

= 0.14912

= 14.912%

c)

P(gas problem | negative test result)

= 45/8290

= 0.005428

= 0.543 %