Question

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 0.5 below the number to 0.5 above the number. For example, approximate a binomial probability P(≥10) by finding the Normal probability P(≥9.5) . Be careful: binomial P(>10) is approximated by Normal P(≥10.5) . According to CDC estimates, more than 2 million people in the United States are sickened each year with antibiotic‑resistant infections, with at least 23,000 dying as a result. Antibiotic resistance occurs when disease‑causing 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 800,000 cases occurring annually, with approximately 30% of those cases resistant to any antibiotic. Suppose a local health clinic sees 40 cases. The exact binomial probability that 15 or more cases are resistant to any antibiotic is 0.193 .

(a) Show that this setting satisfies the rule of thumb for the use of the Normal approximation (just barely).

Answer 1

The Normal approximation to the binomial distribution is appropriate in the given scenario, as both np (12) and nq (28) exceed the minimum value of 5 as per the rule of thumb.

Step-by-step explanation:

To demonstrate that the use of the Normal approximation to the binomial distribution is justified in the given scenario, the standard rule of thumb must be examined. The rule of thumb suggests that the Normal approximation can be used for a binomial distribution when both np and nq are greater than five. In our case, with n being the number of trials (40) and p being the probability of success (30% of cases resistant), we can calculate these values as follows:

• np = 40 trials * 30% = 12 successes
• nq = 40 trials * 70% = 28 failures

Since both np and nq are greater than 5, the use of the Normal approximation is appropriate for the binomial distribution of antibiotic-resistant gonorrhea cases. This setting narrowly satisfies the rule of thumb, as np is not quite 10, but it is still acceptable.