A survey conducted in 2011 asked employed adults from age 25 to 70 if they plan to keep a foot in the working world after retirement. Thirty-three percent of the respondents answered "very likely". A random sample of 75 adults from this age group was asked the same question. Using the normal approximation to the binomial distribution, what is the probability that 20, 21, 22, or 23 adults from this sample would very likely plan to keep a foot in the working world after retirement?

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The probability of an individual planning to keep a foot in the working world after retirement is 0.33 (33%). For a sample size of 75, the mean (μ) of the distribution is n*p = 75*0.33 = 24.75, and the standard deviation (σ) is sqrt(n*p*(1-p)) = sqrt(75*0.33*0.67) ≈ 4.27. To find the probability of having 20, 21, 22, or 23 adults planning to keep working, we use the normal approximation to the binomial distribution. We calculate the z-scores for 19.5 (since we're looking for more than 20) and 23.5 (for less than 24) and then find the area under the normal curve between these z-scores. The z-score for 19.5 is (19.5-24.75)/4.27 ≈ -1.23, and for 23.5, it is (23.5-24.75)/4.27 ≈ -0.29. Looking these up in a z-table or using a calculator gives us the probabilities for each, and the difference between them gives the probability of having between 20 and 23 inclusive adults planning to keep working, which is approximately 0.3821.

A Survey Conducted in 2011 Asked Employed Adults from Age