If an average person listens to music for 1,000 hours in a year, the probability that the component lasts for more than 3 years is 2.3%.
In Mathematics and Statistics, the z-score of a given sample size or data set can be calculated by using the following formula:
Z-score, z = (x - μ)/σ
Where:
σ represents the standard deviation.
x represents the sample score.
μ represents the mean score.
Since an average person listens to music for 1,000 hours in a year, the total life span of the component for 3 years can be calculated as follows;
Total life span = 3 × 1000
Total life span = 3000 years.
By substituting the parameters into the z-score formula, we have the following:
Z-score, z = (3000 - 2400)/300
Z-score, z = 2.0
Based on the standardized normal distribution table, the required probability is given by:
P(X > 3000) = P(x > Z)
P(X > 3000) = 1 - P(Z < 2)
P(X > 3000) = 1 - 0.9773
Probability = 0.0227 × 100.
Probability = 2.27 ≈ 2.3%.