Assume that the age for first occurrence of filing personal taxes follows a roughly normal distribution, with a mean $(\mu)$ of 22 years and a standard deviation $(\sigma)$ of 5 years.
What age defines the lowest $40 \%$ of ages?
$18.6 \%$
23.3
25.4
20.7
Answer
Final Answer: The age that defines the lowest 40% of ages is approximately \(\boxed{20.73}\) years.
Step 1 ：Assume that the age for first occurrence of filing personal taxes follows a roughly normal distribution, with a mean $(\mu)$ of 22 years and a standard deviation $(\sigma)$ of 5 years.
Step 2 ：We are asked to find the age that defines the lowest 40% of ages. This is a problem of finding the percentile of a normal distribution. The 40th percentile (or the lowest 40% of ages) is the age below which 40% of the observations fall.
Step 3 ：We can use the z-score formula to solve this problem. The z-score is the number of standard deviations a particular score is from the mean. In this case, we need to find the z-score that corresponds to the 40th percentile.
Step 4 ：Using the z-score formula, we find that the z-score is -0.2533471031357997.
Step 5 ：We then use this z-score to find the corresponding age. The formula for this is age = \(\mu + z \times \sigma\). Substituting the given values, we get age = 22 + (-0.2533471031357997) * 5 = 20.733264484321.
Step 6 ：This means that the lowest 40% of ages for first occurrence of filing personal taxes is around 20.73 years.
Step 7 ：Final Answer: The age that defines the lowest 40% of ages is approximately \(\boxed{20.73}\) years.
