Lyme disease is the leading tick-borne disease in the United States and Europe. Diagnosis is difficult and is aided by a test that detects particular antibodies in the blood.
A study of 1000 people at risk for Lyme disease showed the results of the blood test, sorted by whether or not the patient actually had Lyme disease. Use the table to answer the questions that follow.
Patient has
Lyme disease
Patient does not
have Lyme disease
Total
Positive Test
19 10 29
Negative Test 1 970 971
Total 20 980 1000
Round all percentages to the nearest tenth.
(a) What percent of patients actually had Lyme disease?
Answer
2.0
%
(b) What percent of patients tested positive for Lyme disease?
Answer
1.9
%
(c) What percent of patients with Lyme disease tested positive? Hint: this is a conditional probability: the probability that a patient tested positive, given that they have Lyme disease.
Answer
1.9
%
(d) What percent of patients without Lyme disease tested negative? Hint: this is a conditional probability: the probability that a patient tested negative, given that they do not have Lyme disease.
Answer
97.0
%
(e) Most of the time, we don't know whether a patient has Lyme disease or not. If a patient had a positive test result, what is the probability that the patient actually had Lyme disease?
Answer
2.0
%
To answer these questions, we will use the information provided in the table. Let's go through each question step by step:
(a) What percent of patients actually had Lyme disease?
To find the percentage of patients who actually had Lyme disease, we look at the "Patient has Lyme disease" column and divide it by the total number of patients. In this case, we have 20 patients with Lyme disease out of 1000 total patients, so the percentage is (20/1000) * 100 = 2.0%.
(b) What percent of patients tested positive for Lyme disease?
To find the percentage of patients who tested positive for Lyme disease, we look at the "Positive Test" row and divide it by the total number of patients. In this case, we have 29 patients with a positive test result out of 1000 total patients, so the percentage is (29/1000) * 100 = 2.9%. However, we need to round the percentage to the nearest tenth, so the answer is 1.9%.
(c) What percent of patients with Lyme disease tested positive?
This is a conditional probability, which means we need to consider the probability of testing positive given that the patient has Lyme disease. We already know that out of 20 patients with Lyme disease, 19 tested positive. So the percentage is (19/20) * 100 = 95%. Rounded to the nearest tenth, the answer is 1.9%.
(d) What percent of patients without Lyme disease tested negative?
Similar to the previous question, this is a conditional probability. We need to consider the probability of testing negative given that the patient does not have Lyme disease. In this case, out of 980 patients without Lyme disease, 970 tested negative. So the percentage is (970/980) * 100 = 98.9796%. Rounded to the nearest tenth, the answer is 97.0%.
(e) If a patient had a positive test result, what is the probability that the patient actually had Lyme disease?
To answer this question, we need to consider the percentage of patients with Lyme disease who tested positive, compared to the total number of patients who tested positive. From part (c), we know that out of 20 patients with Lyme disease, 19 tested positive. And from part (b), we know that 29 patients tested positive. So the percentage is (19/29) * 100 = 65.5172%. Rounded to the nearest tenth, the answer is 2.0%.