Question

use the following results from a test for marijuana use, which is provided by a certain drug testing company. among 141 subjects with positive test results, there are 20 false positive results. among 153 negative results, there are 2 false negative results. complete parts (a) through (c) (hint: construct a table.)
a. how many subjects were included in the study?
the total number of subjects in the study was 294
b. how many subjects did not use marijuana?
a total of \\(\square\\) subjects did not use marijuana

Explanation:

Step1: Understand the problem

We know that there are 153 negative test results, and among them, 2 are false negatives. False negatives are subjects who actually used marijuana but tested negative. So the number of subjects who did not use marijuana and tested negative is the total negative results minus false negatives. But actually, the total number of subjects who did not use marijuana is the number of true negatives plus false positives? Wait, no. Wait, the negative test results: true negatives (did not use, tested negative) and false negatives (used, tested negative). So the number of subjects who did not use marijuana is the number of true negatives plus false positives? Wait, no. Wait, let's think again.

Wait, the negative test results: 153. Among these, 2 are false negatives (so they used marijuana but tested negative). So the number of subjects who did not use marijuana and tested negative (true negatives) is 153 - 2 = 151? Wait, no, that's not right. Wait, the false positives are among the positive test results. Wait, the problem is about subjects who did not use marijuana. So subjects who did not use marijuana can have either a negative test result (true negative) or a positive test result (false positive).

So the number of subjects who did not use marijuana is (number of true negatives) + (number of false positives).

We know that the number of positive test results is 141, with 20 false positives (so false positives are subjects who did not use marijuana but tested positive). The number of negative test results is 153, with 2 false negatives (so false negatives are subjects who used marijuana but tested negative). So the number of true negatives (did not use, tested negative) is 153 - 2 = 151? Wait, no, the negative test results: total negative is 153. False negatives are 2 (used, tested negative), so true negatives (did not use, tested negative) is 153 - 2 = 151? Wait, no, that's incorrect. Wait, the negative test results are either true negatives (did not use) or false negatives (used). So true negatives = negative results - false negatives = 153 - 2 = 151. Then, the false positives are 20 (did not use, tested positive). So total subjects who did not use marijuana is true negatives + false positives = 151 + 20 = 171? Wait, let's check:

Wait, total positive results: 141. False positives: 20, so true positives (used, tested positive) = 141 - 20 = 121.

Total negative results: 153. False negatives: 2, so true negatives (did not use, tested negative) = 153 - 2 = 151.

Then, total subjects who used marijuana: true positives + false negatives = 121 + 2 = 123.

Total subjects who did not use marijuana: true negatives + false positives = 151 + 20 = 171.

Alternatively, total subjects in the study is 141 (positive) + 153 (negative) = 294, which matches part (a). Then, total subjects who did not use marijuana is total subjects - total subjects who used marijuana. Total subjects who used marijuana: true positives (141 - 20 = 121) + false negatives (2) = 123. So 294 - 123 = 171. Yes, that works.

So step by step:

Step1: Find total positive and negative results

Total positive results: 141, total negative results: 153. Total subjects: 141 + 153 = 294 (from part a).

Step2: Find number of subjects who used marijuana

True positives (used, tested positive) = total positive - false positives = 141 - 20 = 121.

False negatives (used, tested negative) = 2.

So total who used marijuana = 121 + 2 = 123.

Step3: Find number of subjects who did not use marijuana

Total subjects - total who used marijuana = 294 - 123 = 171.

Alternatively, true negatives (did not use, tested negative) = total negative - false negatives = 153 - 2 = 151.

False positives (did not use, tested positive) = 20.

So total who did not use = 151 + 20 = 171.

Answer:

171