Final answer:
To calculate the probability that a duxcell battery will last longer than an infinitycell battery, compare the lifetimes of the two types of batteries using their means and standard deviations. The probability is approximately 0.3918.
Step-by-step explanation:
To find the probability that a duxcell battery will last longer than an infinitycell battery, we need to compare the lifetimes of the two types of batteries. Given that the mean and standard deviation of the lifetimes of duxcell batteries are 10 and 2 minutes respectively, and the mean and standard deviation for the infinitycell batteries are 11 and 3 minutes respectively, we can use the normal distribution to calculate the probability.
We know that the difference in means between the two types of batteries is 10 - 11 = -1. To compare the lifetimes, we need to calculate the z-score of the difference in means. The z-score is calculated as (x - μ) / σ, where x is the value we want to compare, μ is the mean, and σ is the standard deviation. In this case, the z-score is (-1 - 0) / √(2^2 + 3^2) = -1 / √13 ≈ -0.2774.
Using the Z-table, we can find the probability associated with the z-score.
The probability of a duxcell battery lasting longer than an infinitycell battery is the same as the probability of having a negative z-score. Looking up the z-score in the table, we find that the probability is approximately 0.3918.