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 from 0.50.5 below the number to 0.50.5 above the number. For example, approximate a binomial probability (10)P(X10) by finding the Normal probability (9.5).P(X9.5). Be careful: binomial (>10)P(X>10) is approximated by Normal(10.5).NormalP(X10.5).

According to CDC estimates, at least 2.8million2.8million people in the United States are sickened each year with antibioticresistant infections, and at least 35,00035,000 die 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 estimated 1.13million1.13million new cases occurring annually, and approximately 50%50% of those cases are resistant to anyantibiotic.

Suppose a local health clinic sees 20cases.20cases. The exact binomial probability that 1313 or more cases are resistant to any antibiotic is0.1316.is0.1316.

(a) Show that this setting satisfies the rule of thumb for the use of the Normal approximation (just barely). Give your answers to one decimalplace.

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