One reason the Normal approximation may fail to give accurate estimates of binomial probabilities is that the binomial distributions are discrete and the Normal distributions are continuous. That is, counts take only whole number values, but Normal variables can take any value. We can improve the Normal approximation by treating each whole number count as if it occupied the interval from0.5below the number to0.5above the number. For example, approximate a binomial probability(10)by finding the Normal probability(9.5).Be careful: binomial(>10)is approximated byNormal (10.5).

According to CDC estimates, at least2.8 million people in the United States are sickened each year with antibioticresistant infections, and at least35,000die as a result. Antibiotic resistance occurs when diseasecausing microbes become resistant to antibiotic drug therapy. Because this resistance is typically genetic and transferred to the next generations of microbes, it is a very serious public health problem. Of the three infections considered most serious by the CDC, gonorrhea has an estimated1.13 million new cases occurring annually, and approximately50%of those cases are resistant to any antibiotic.

Suppose a local health clinic sees20 cases.The exact binomial probability that13or more cases are resistant to any antibiotic is 0.1316.

1. What is the Normal approximation to(13)?Give your answer to four decimal places.

2.What is the Normal approximation using the continuity correction? That's a lot closer to the true binomial probability. Give your answer to four decimal places.

(13)=