Answer :
To ensure that no more than 5% of cell phone batteries require charging in 13 hours or less, the mean time between charges must be approximately 15.47 hours.
To determine the mean time between charges such that no more than 5% of the cell phone batteries require charging in 13 hours or less, we need to find the 5th percentile of the normal distribution. Given that the standard deviation is 1.5 hours, we can use the z-score formula and standard normal distribution tables or a calculator with normal distribution functions.
First, we need to find the z-score that corresponds to the 5th percentile. We can reference a standard normal distribution table or use a calculator to find that the z-score for the 5th percentile is approximately -1.645.
The z-score formula is: Z = (X - mean) / standard deviation
Where X is the value (13 hours), Z is the z-score (-1.645), and the standard deviation is 1.5 hours.
We can rearrange this formula to solve for the mean: mean = X - (Z * standard deviation) = 13 - (-1.645 * 1.5) = 13 + 2.4675 = 15.4675 hours
Therefore, the mean time between charges must be approximately 15.47 hours to ensure that no more than 5% of the batteries require charging in 13 hours or less.