The amount of time a certain brand of light bulb lasts is normally distributed with a mean of 1900 hours and a standard deviation of 60 hours. What is the probability that a randomly chosen light bulb will last between 1820 hours and 2030 hours, to the nearest thousandth ?
To solve this problem, we can first standardize the values using the z-score formula:
z = (X - μ) / σ
For X = 1820 hours:
z1 = (1820 - 1900) / 60 = -80 / 60 = -1.3333
For X = 2030 hours:
z2 = (2030 - 1900) / 60 = 130 / 60 = 2.1667
Next, we can use a Z-table or a calculator to find the probabilities associated with these z-scores:
P(-1.3333 < Z < 2.1667) = P(Z < 2.1667) - P(Z < -1.3333)
Using a Z-table, we find:
P(Z < 2.1667) ≈ 0.9857
P(Z < -1.3333) ≈ 0.0912
Therefore, the probability that a randomly chosen light bulb will last between 1820 hours and 2030 hours is approximately:
0.9857 - 0.0912 = 0.8945
So, the probability that a randomly chosen light bulb will last between 1820 hours and 2030 hours is approximately 0.895.