More than a decade ago, high levels of lead in the blood put 85% of children at risk. A concerted effort was made to remove lead from the environment. Now, suppose only 16% of children in the United States are at risk of high blood-lead levels.

(a) In a random sample of 184 children taken more than a decade ago, what is the probability that 50 or more had high blood-lead levels? (Round your answer to three decimal places.)

1 = 100% probability that 50 or more had high blood-lead levels

Step-by-step explanation:

I am going to use the binomial approximation to the normal to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

The standard deviation of the binomial distribution is:

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that , .

In this problem, we have that:

p is 0.85 because we are working with a sample from a decade ago.

So

(a) In a random sample of 184 children taken more than a decade ago, what is the probability that 50 or more had high blood-lead levels? (Round your answer to three decimal places.)

This is 1 subtracted by the pvalue of Z when X = 50. So

has a pvalue of 0.

1 - 0 = 1

1 = 100% probability that 50 or more had high blood-lead levels