To solve this problem, we can use the normal approximation to the binomial distribution. We're given that the probability \( p \) of an adult smoker acquiring the habit by age 14 is \( p = \frac{2}{5} = 0.4 \). We have a sample size of \( n = 400 \).
Step 1: Calculate the mean and standard deviation
The mean \( \mu \) of a binomial distribution is given by:
\[ \mu = n \cdot p = 400 \cdot 0.4 = 160 \]
The standard deviation \( \sigma \) is calculated as follows:
\[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{400 \cdot 0.4 \cdot 0.6} = \sqrt{400 \cdot 0.24} = \sqrt{96} \approx 9.798 \]
Step 2: Convert the binomial to a normal distribution
We want to find the probability that 170 or more smokers acquired the habit by age 14, or \( P(X \geq 170) \). To use the normal approximation, we will apply a continuity correction. Thus, we find \( P(X \geq 170) \) as \( P(X \geq 169.5) \).
Step 3: Standardize the variable
We convert the binomial random variable \( X \) to a standard normal variable \( Z \):
\[ Z = \frac{X - \mu}{\sigma} = \frac{169.5 - 160}{9.798} \approx \frac{9.5}{9.798} \approx 0.968 \]
Step 4: Find the probability using the standard normal distribution
Now, we look for \( P(Z \geq 0.968) \). We can find this using the standard normal distribution table or a calculator.
The cumulative probability for \( Z = 0.968 \):
\[ P(Z < 0.968) \approx 0.8336 \]
Thus,
\[ P(Z \geq 0.968) = 1 - P(Z < 0.968) \approx 1 - 0.8336 = 0.1664 \]
Final result
Rounding to the nearest thousandth, the probability that 170 or more smokers acquired the habit by age 14 is:
\[ \boxed{0.166} \]
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