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Involve the method of composite sampling, whereby a medical testing laboratory saves time and money by combining blood samples for tests so that only one test is conducted for several people. A combined sample tests positive if at least one person has the disease. If a combined sample tests positive, then individual blood tests are used to identify the individual with the disease or disorder. Based on data from Bloodjournal.org, 10\% of women 65 years of age and older have anemia, which is a deficiency of red blood cells. In tests for anemia, blood samples from 8 women 65 and older are combined. What is the probability that the combined sample tests positive for anemia? Is it likely for such a combined sample to test positive?

Step by step solution

01

- Identify Probability of Individual

Establish the probability that a single woman 65 years of age or older has anemia. According to the problem, this probability is 10%, or 0.1.

02

- Calculate Probability of No Anemia in Single Woman

The probability that a given woman does not have anemia is the complement of the probability that she does have it. Therefore, it is 1 - 0.1 = 0.9.

03

- Calculate Probability of No Anemia in Combined Sample

To determine the probability that a combined sample from 8 women tests negative (meaning none of them has anemia), we calculate the probability that all 8 women do not have anemia. This is done by raising the probability of no anemia in a single woman to the 8th power: 0.9^8.

04

- Compute the Value

Calculate the value: \[0.9^8 \approx 0.4305.\]

05

- Calculate Probability of Positive Test

To find the probability that the combined sample tests positive, subtract the probability of a negative test from 1: \[1 - 0.4305 = 0.5695.\]

06

- Assess Likelihood

Since we find that the probability of a combined sample testing positive is around 57%, it is fairly likely for such a combined sample to test positive for anemia.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

anemia probability calculation

To understand the probability calculation for anemia in a combined blood sample, we first need to grasp how individual probabilities work. Anemia affects 10% of women aged 65 and older. This means the probability that one woman in this age group has anemia is \(0.1\). Conversely, the probability that a single woman does not have anemia is \(1 - 0.1 = 0.9\).

combined blood sample tests

Composite sampling simplifies medical testing by combining blood samples. If any person has a condition, the combined sample tests positive. For anemia, when 8 women’s samples are combined, we need to calculate the chance that none have anemia to determine the combined sample's test result.
The probability of each woman not having anemia is 0.9. We raise this probability to the power of the number of women (8), resulting in \(0.9^8\). Calculating this, we find \(0.9^8 ≈ 0.4305\).

step-by-step probability solution

To determine the overall probability for a combined blood sample, follow these steps:

• Identify the individual probability for anemia in one woman: \(0.1\).
• Determine the probability that one woman does not have anemia: \(1 - 0.1 = 0.9\).
• Calculate the probability that none of the 8 women have anemia: \[0.9^8 ≈ 0.4305\].

Finally, subtract this from 1 to find the probability of a positive test: \(1 - 0.4305 = 0.5695\). The likelihood of a positive test for anemia in this group is around 57%, indicating it's fairly likely for such a combined sample to test positive.