Question

a new car battery is sold with a two-year warranty whereby the owner gets the battery replaced free of cost if it breaks down during the warranty period. suppose an auto store makes a net profit of $20 on batteries that stay trouble-free during the warranty period; it makes a net loss of $10 on batteries that break down. the life of batteries is known to be normally distributed with a mean and a standard deviation of 40 months and 16 months, respectively. [you may find it useful to reference the z table.] a. what is the probability that a battery will break down during the warranty period?

Answer 1

The probability that a battery will break down during the warranty period can be calculated using the normal distribution and z-scores.

Step-by-step explanation:

The probability that a battery will break down during the warranty period can be determined using the normal distribution and z-scores.

First, we need to calculate the z-score for the value of 24 months (warranty period) using the formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.

Plugging in the values: z = (24 - 40) / 16 = -1

Using the z-table, the probability of a battery breaking down during the warranty period (z-score of -1) is approximately 0.1587 or 15.87%.

Answer 2

There is approximately a 15.87% chance that a battery will fail within the two-year warranty period, based on the normal distribution with a given mean of 40 months and a standard deviation of 16 months.

Step-by-step explanation:

The life of car batteries is known to be normally distributed with a mean of 40 months and a standard deviation of 16 months. To calculate the probability that a battery will break down during the two-year warranty period (24 months), we can use the z-score formula:

Z = (X - μ) / σ

Where X is the value we are interested in (24 months), μ is the mean (40 months), and σ is the standard deviation (16 months). Plugging in the numbers we get:

Z = (24 - 40) / 16 = -1

Consulting a z-table, a Z score of -1 corresponds to approximately 0.1587 (15.87%). Therefore, there is about a 15.87% chance that a battery will fail within the two-year warranty period.