Question

The lifetimes of two brands of light bulbs, A and B, are normally distributed. Brand A has a mean lifetime of 800 hours with a standard deviation of 50 hours, while brand B has a mean lifetime of 850 hours with a standard deviation of 60 hours. If a random light bulb is selected and it lasts more than 820 hours, what is the probability that it belongs to brand B?

Answer 1

To find the probability that a randomly selected light bulb lasting more than 820 hours belongs to brand B, we can use the concept of z-scores and the standard normal distribution.

First, let's calculate the z-score for 820 hours for brand B. The z-score formula is given by:

z = (x - μ) / σ

where x is the value (820 hours), μ is the mean (850 hours), and σ is the standard deviation (60 hours) for brand B.

Calculating the z-score:

z = (820 - 850) / 60 = -0.5

Now, we can find the probability using the standard normal distribution table or a calculator. The probability of a z-score less than -0.5 is approximately 0.3085.

However, we want the probability that the light bulb lasts more than 820 hours, so we subtract the probability from 1:

P(light bulb belongs to brand B) = 1 - 0.3085 = 0.6915

Therefore, the probability that the light bulb belongs to brand B is approximately 0.6915 or 69.15%.