The life of an electric light bulb is known to be normally distributed with a mean of 2000 hours and a std. of 120 hours. Find the probability that the life of such a bulb would be.
(a) greater than 2150 hours
(b) within the range of 1850 hours to 2090 hours
0.7768
To find the probability in each scenario, we can use the concept of the standard normal distribution, which allows us to convert any normal distribution into a standard normal distribution with a mean of 0 and a standard deviation of 1.
1. To find the probability that the life of the bulb is greater than 2150 hours:
(a) Standardize the given value using z-score formula: z = (X - μ) / σ
X = 2150 hours, μ = 2000 hours, σ = 120 hours
z = (2150 - 2000) / 120 = 1.25
(b) Use the standard normal distribution table or a calculator to find the probability (P(Z > 1.25)).
The table will give you the probability corresponding to the z-score 1.25 (P(Z > 1.25)).
2. To find the probability that the life of the bulb is within the range of 1850 hours to 2090 hours:
(a) Standardize the lower and upper bounds using z-score formula:
For the lower bound (1850 hours):
z_lower = (1850 - 2000) / 120 = -1.25
For the upper bound (2090 hours):
z_upper = (2090 - 2000) / 120 = 0.75
(b) Use the standard normal distribution table or a calculator to find the probabilities (P(Z < -1.25) and P(Z < 0.75)).
The table will give you the probabilities corresponding to the z-scores -1.25 (P(Z < -1.25)) and 0.75 (P(Z < 0.75)).
Finally, subtract P(Z < -1.25) from P(Z < 0.75) to get the probability of being within this range (P(-1.25 < Z < 0.75)).
Note: If you're using a calculator, most scientific calculators can directly provide probabilities for standard normal distribution values. For example, using the "normalcdf" function in many scientific calculators, you can find P(Z > 1.25), P(Z < -1.25), and P(-1.25 < Z < 0.75) directly.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
a. in smaller portion
b. Between Z scores and mean