Final answer:
The marketing department should guarantee a lifespan of approximately 131.8 hours to ensure that 98% of the batteries meet the guarantee, based on the normal distribution with a mean of 140 hours and a standard deviation of 4 hours.
Step-by-step explanation:
The mean lifespan of the batteries is given as 140 hours with a standard deviation of 4 hours. The marketing department wants to guarantee a lifespan that 98% of the batteries will meet. To determine this lifespan, we use the concept of the normal distribution and z-scores.
Since the normal distribution is symmetric around the mean, we look for the lifespan corresponding to the 2nd percentile (z-score for 0.02). This is because 100% - 98% = 2%, and we are concerned with the lower end of the distribution where batteries will not meet the guaranteed life.
To find the exact lifespan, we can use the z-score formula:
z = (X - mean) / standard deviation. Looking up the z-score for 0.02 in z-tables or using a statistical software, we get approximately <-strong>2.05. Solving for X gives us the guaranteed lifespan:
X = mean + (z × standard deviation)
X = 140 + (-2.05 × 4)
X ≈ 131.8 hours.
Therefore, the marketing department should guarantee that the batteries will last at least 131.8 hours to ensure that 98% of the batteries meet the guarantee.