Answer:
Approximately 6.05 years
Step-by-step explanation:
Use the properties of the normal distribution
Given:
Mean() = 7 years
Variance (^2) = \(2 years^{2}\)
Standard deviation() = \(\sqrt{2}\) years
Find the value of x such that 25% of the distribution lies below x and 75% lies above x.
The z-score z for the 25th percentile is approximately -0.674.
The relationship between the z-score and the value x in the normal distribution is given by the formula:
x = μ + z ⋅ σ
Substitute the given values
x = 7 + (-0.674) \(\sqrt{2}\)
Calculate for x
\(\sqrt{2}\) = approx 1.414
x = 7 + (-0.674) 1.414
x = 7 - 0.953
x = approx 6.047
Therefore, the replacement time separating the bottom 25% from the top 75% is approximately 6.05 years.