Final answer:
To ensure there is no more than a 9% chance of a phone dying, you should plan to recharge the phone after approximately 15.3 hours. This calculation is based on the normal distribution with a mean of 11.25 hours and a standard deviation of 3.1 hours.
Step-by-step explanation:
The question requires the application of normal distribution principles to find the recharging schedule for a phone's battery life. According to the information provided, the mean battery life is 11.25 hours with a standard deviation of 3.1 hours. The goal is to determine the time to recharge the phone so that there is no more than a 9% chance that the battery will die.
Firstly, we need to find the z-score that corresponds to the 91st percentile since we want the battery to last longer than 91% of the time, leaving a 9% chance of it dying. Using a z-table or normal distribution calculator, we find that the z-score for 0.91 is approximately 1.34. Next, we use Z = (X - μ) / σ, where Z is the z-score, X is the time after which we want to recharge, μ is the mean, and σ is the standard deviation.
With the z-score and other values, we formulate the equation 1.34 = (X - 11.25) / 3.1. Solving for X gives us the time to recharge: X = 1.34 * 3.1 + 11.25, which after calculation provides us with a time of approximately 15.3 hours. Therefore, you should plan to recharge your phone after 15.3 hours to ensure there is no more than a 9% chance it will die.