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.

- Label the cells as true positives, false positives, false negatives, true negatives and Type I and Type II errors.
- Complete the matrix to show the values for true positives, false positives (Type I), false negatives (Type II), and true negatives
- 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 %

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