We can solve this problem using the z-score formula:
z = (X - μ) / (σ / sqrt(n))
Where:
X = sample mean
μ = population mean
σ = population standard deviation
n = sample size
Given:
X = unknown
μ = 1600 hours
σ = 400 hours
n = 64
probability (P) = 0.1492
We need to find the z-score corresponding to a probability of 0.1492, which is -1.0301 (found using a z-table or calculator). Now we can substitute the values into the z-score formula to solve for X:
-1.0301 = (X - 1600) / (400 / sqrt(64))
-1.0301 = (X - 1600) / (400 / 8)
-1.0301 = (X - 1600) / 50
-51.505 = X - 1600
X = 1600 - 51.505
X ≈ 1548.495
Therefore, the sample mean lifetime is more than 1548 hours with a probability of 0.1492.
Out of the options provided:
1580 hours is the closest value to 1548.495. So, the answer is 1580.
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