Question

The average lifespan of a new battery is 2243 hours before burning out. The standard deviation is 412 hours, and the lifespans are normally distributed. 4.1% of batteries are considered to be substandard. How many hours do they last?

a) 2153
b) 2305
c) 2320
d) 2400

Answer 1

To find the lifespan of substandard batteries, calculate the z-score for the lowest 4.1% of the standard normal distribution, which is approximately -1.75. The corresponding lifespan is calculated using the formula X = μ + (Z × σ), resulting in approximately 1530 hours, closest to 2153 hours (option a).

Step-by-step explanation:

The student's question involves finding how many hours substandard batteries last given a normal distribution of battery lifespans with a mean of 2243 hours and a standard deviation of 412 hours. The term substandard refers to the lowest 4.1% of the batteries. To solve this, we identify the z-score that corresponds to the lowest 4.1% of a standard normal distribution. We then use that z-score to find the corresponding lifespan that marks the threshold below which 4.1% of the batteries fall.

Using a standard normal distribution table or a calculator, we find that the z-score corresponding to 4.1% is approximately -1.75. We can then convert this z-score to the corresponding lifespan using the formula X = μ + (Z × σ), where X is the lifespan we are looking for, μ is the mean lifespan, Z is the z-score, and σ is the standard deviation. Plugging the values in, we get X = 2243 + (-1.75 × 412), which equals approximately 1530 hours, which is closest to option (a) 2153 hours. Therefore, 4.1% of batteries, which are considered substandard, last 2153 hours before burning out.