The lifetimes of a certain brand of light bulbs are known to be normally distributed with a mean of 1600 hours and a standard deviation of 400 hours. A random sample of 64 of these light bulbs is taken. The probability is 0.1492 that the sample mean lifetime is more than how many hours?
1652
1500
1724
1580
1578
To solve this problem, we need to calculate the z-score corresponding to a probability of 0.1492 and then use it to find the corresponding sample mean lifetime.
First, we find the z-score using the formula:
z = (X - μ) / (σ / sqrt(n))
Where:
X = sample mean
μ = population mean
σ = population standard deviation
n = sample size
Given:
μ = 1600
σ = 400
n = 64
P(X > x) = 0.1492
We can calculate the z-score as:
z = (X - 1600) / (400 / sqrt(64))
z = (X - 1600) / 50
Next, we look up the z-score in a standard normal distribution table or calculator to find the value that corresponds to a cumulative probability of 0.8518 (1 - 0.1492).
In the standard normal distribution table, we find that the z-score is approximately 1.03.
Now we can find the sample mean lifetime for the given probability:
1.03 = (X - 1600) / 50
Solving for X, we get:
X = 1.03 * 50 + 1600
X ≈ 1652
Therefore, the probability is 0.1492 that the sample mean lifetime is more than 1652 hours.