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.
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
1 answer