More than three decades ago, high levels of lead in the...

More than three decades ago, high levels of lead in the blood put $88 \%$ of children at risk. A concerted effort was made to remove lead from the environment. Now, according to the Third National Health and Nutrition Examination Survey (NHANES III) conducted by the Centers for Disease Control and Prevention, only $9 \%$ of children in the United States are at risk of high blood-lead levels. (a) In a random sample of 200 children taken more than three decades ago, what is the probability that 50 or more had high blood-lead levels? (b) In a random sample of 200 children taken now, what is the probability that 50 or more have high blood-lead levels?

More than three decades ago, high levels of lead in the blood put $88 \%$ of children at risk. A concerted effort was made to remove lead from the environment. Now, according to the Third National Health and Nutrition Examination Survey (NHANES III) conducted by the Centers for Disease Control and Prevention, only $9 \%$ of children in the United States are at risk of high blood-lead levels.
(a) In a random sample of 200 children taken more than three decades ago, what is the probability that 50 or more had high blood-lead levels?
(b) In a random sample of 200 children taken now, what is the probability that 50 or more have high
blood-lead levels?

Key Concepts

Normal Approximation to the Binomial

When the number of trials is large and the probability of success is neither very close to 0 nor 1, the binomial distribution can be approximated by a normal distribution. This is based on the Central Limit Theorem, which allows easier computation of probabilities by using the normal distribution with a mean equal to np and a variance equal to np(1-p). In this problem, such an approximation can simplify finding the probability that 50 or more children have high blood-lead levels.

Standardization (Z-Score)

Standardization involves converting a value from a normal distribution to a Z-score, which represents the number of standard deviations away from the mean. This process is used to find probabilities by referencing the standard normal distribution, enabling the calculation of the probability that a binomially distributed variable (approximated as normal) exceeds a certain threshold.

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. In the context of this question, each child being at risk or not represents a trial with a fixed probability (either 88% in the past or 9% now), and the number of children with high blood-lead levels follows a binomial distribution.

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