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. What is the probability that the sample mean lifetime is more than 1550 hours?
0.9616
0.9242
0.8413
0.7686
To solve this problem, we can calculate the z-score for a sample mean of 1550 hours using the formula:
z = (X̄ - μ) / (σ / √n)
where:
X̄ = sample mean (1550 hours)
μ = population mean (1600 hours)
σ = population standard deviation (400 hours)
n = sample size (64 bulbs)
Plugging in the values:
z = (1550 - 1600) / (400 / √64)
z = -50 / 50
z = -1
Now, we look up the z-score of -1 in a standard normal distribution table to find the probability that the sample mean lifetime is more than 1550 hours. The area to the left of z = -1 is 0.1587, so the probability to the right (more than 1550 hours) is 1 - 0.1587 = 0.8413.
Therefore, the probability that the sample mean lifetime is more than 1550 hours is 0.8413.
Therefore, answer is:
0.8413