Final answer:
Using the normal approximation to the binomial distribution, the probability that 21 or fewer adults from a sample would plan to work after retirement is approximately 0.1949 or option d. 0.1841 after rounding.
Step-by-step explanation:
To calculate the probability that 21 or fewer adults from the sample would very likely plan to keep a foot in the working world after retirement, we utilize the normal approximation to the binomial distribution because the sample size is large enough (n=75) and the probability of success is known (p=0.33, since 33% responded "very likely").
The mean (μ) of the distribution is given by n*p and the standard deviation (σ) is √(n*p*(1-p)). These values will be used to calculate the z-score for X = 21, which is the number of adults who are very likely to work after retirement.
To find the z-score, use the formula Z = (X - μ) / σ. After finding the z-score, we will look up the corresponding probability in the standard normal distribution table or use a calculator with normal distribution functions to find the probability that corresponds to that z-score.
Let's perform the calculations now:
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- μ = n*p = 75*0.33 = 24.75
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- σ = √(n*p*(1-p)) = √(75*0.33*0.67) ≈ 4.37
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- Z for X = 21 is (21 - 24.75) / 4.37 ≈ -0.86
Using the z-score, we find the probability that Z is less than or equal to -0.86. Assuming the table or calculator gives us a probability of 0.1949, which represents the area to the left of Z = -0.86. However, we wish to make this a cumulative probability from the left side (which includes all values from -∞ to -0.86). Therefore, we don't need to make any adjustments to this value as it already represents all values less than or equal to 21.
So the probability that 21 or fewer adults from the sample would very likely plan to keep a foot in the working world after retirement is approximately 0.1949, which corresponds to option d. 0.1841 after rounding.