Question \( 4[3+3+3=9 \) marks \( ] \) A company manufacturing car batteries finds the battery's life in years to be normally distributed with \( \mu=4 \) and \( \sigma=1 \) year. What is the probability that a car battery will have a life? a) between 3 and 5 years b) longer than 6 years c) less than 1 year

To solve the problem, we will use the properties of the normal distribution. Given that the battery life is normally distributed with a mean (\( \mu \)) of 4 years and a standard deviation (\( \sigma \)) of 1 year, we can standardize the values using the Z-score formula: \[ Z = \frac{X - \mu}{\sigma} \] where \( X \) is the value for which we want to find the probability. ### a) Probability that a car battery will have a life between 3 and 5 years 1. **Calculate the Z-scores for 3 and 5 years:** - For \( X = 3 \): \[ Z_3 = \frac{3 - 4}{1} = -1 \] - For \( X = 5 \): \[ Z_5 = \frac{5 - 4}{1} = 1 \] 2. **Find the probabilities corresponding to these Z-scores using the standard normal distribution table:** - \( P(Z < -1) \) (for \( Z = -1 \)) is approximately 0.1587. - \( P(Z < 1) \) (for \( Z = 1 \)) is approximately 0.8413. 3. **Calculate the probability that the battery life is between 3 and 5 years:** \[ P(3 < X < 5) = P(Z < 1) - P(Z < -1) = 0.8413 - 0.1587 = 0.6826 \] ### b) Probability that a car battery will have a life longer than 6 years 1. **Calculate the Z-score for 6 years:** \[ Z_6 = \frac{6 - 4}{1} = 2 \] 2. **Find the probability corresponding to this Z-score:** - \( P(Z < 2) \) (for \( Z = 2 \)) is approximately 0.9772. 3. **Calculate the probability that the battery life is longer than 6 years:** \[ P(X > 6) = 1 - P(Z < 2) = 1 - 0.9772 = 0.0228 \] ### c) Probability that a car battery will have a life less than 1 year 1. **Calculate the Z-score for 1 year:** \[ Z_1 = \frac{1 - 4}{1} = -3 \] 2. **Find the probability corresponding to this Z-score:** - \( P(Z < -3) \) (for \( Z = -3 \)) is approximately 0.0013. ### Summary of Results - a) Probability that a car battery will have a life between 3 and 5 years: **0.6826** - b) Probability that a car battery will have a life longer than 6 years: **0.0228** - c) Probability that a car battery will have a life less than 1 year: **0.0013**