Answer to Question #256017 in Statistics and Probability for Sheryl

Question

The mean life of a certain brand of auto batteries is 44 months with a standard deviation of 3 months. Assume that the lives of all auto batteries of this brand have ca bell-shaped distribution. Using the empirical rule, find the percentage of auto batteries of this brand that have a life of 41 to 47 months 38 to 50 months 35 to 53 months

Answer 1

$\mu=44 \\ \sigma=3 \\ P(41<X<47) = P(X<47) -P(X<41) \\ = P(Z< \frac{47-44}{3}) -P(Z< \frac{41-44}{3}) \\ = P(Z< 1) -P(Z< -1) \\ = 0.8413 -0.1586 \\ =0.6827 \\ = 68.27 \; \% \\ \\ P(38<X<50) = P(X<50) -P(X<38) \\ = P(Z< \frac{50-44}{3}) -P(Z< \frac{38-44}{3}) \\ =P(Z< 2) -P(Z< -2) \\ = 0.9772 -0.0227 \\ = 0.9545 \\ = 95.45 \; \% \\ \\ P(35<X<53) = P(X<53) -P(X<35) \\ = P(Z < \frac{53-44}{3}) -P(Z< \frac{35-44}{3} ) \\ = P(Z< 3) -P(Z< -3) \\ = 0.9986 -0.0013 \\ = 0.9973 \\ = 99.73 \; \%$