Question

In a survey of adults 80% thought that DNA tests for identifying an individual were very reliable. If you randomly select 100 adults and ask each if he or she thinks that DNA tests for identifying an individual are very reliable. Decide whether you can use normal distribution to approximate the binomial distribution. If so, find the probability that at most 40 people say DNA tests for identifying an individual are very reliable?

Answer 1

4

Explanation:

Sure thing! So, in this case, we can use the normal distribution to approximate the binomial distribution because we have a large enough sample size (100 adults) and the probability of success (80% thinking DNA tests are reliable) is not extremely small or extremely large.

To find the probability that at most 40 people say DNA tests are very reliable, we can use the normal approximation. First, we need to calculate the mean (μ) and standard deviation (σ) of the normal distribution:

μ = n * p = 100 * 0.8 = 80

σ = sqrt(n * p * (1 - p)) = sqrt(100 * 0.8 * 0.2) = 4

Now, we want to find the probability that at most 40 people say DNA tests are very reliable, which means we want to find P(X ≤ 40), where X follows a normal distribution with mean μ = 80 and standard deviation σ = 4.

Using a standard normal distribution table or calculator, you can find P(X ≤ 40). It's about 0.00003167 or approximately 0.0032%. This is the probability that at most 40 people say DNA tests are very reliable when approximating the binomial distribution with a normal distribution.