Home Depot sells compact fluorescent lamps (CFLs) that have a mean life of 10,000 hours with a standard deviation of 1,000 hours. In an order of 8,000 lamps, how many can be expected to last 11,000 hours or longer?
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prob = .1587
multiply that by 8000
1587 x 8000 = 12696000
To solve this problem, we need to use the concept of the standard normal distribution. Here's how we can proceed:
Step 1: Calculate the z-score.
The z-score represents the number of standard deviations an individual data point is from the mean. We can calculate the z-score using the formula:
z = (x - μ) / σ
Where:
- x is the value we are interested in (11,000 hours in this case).
- μ is the mean of the distribution (10,000 hours).
- σ is the standard deviation of the distribution (1,000 hours).
Using the given values, we can substitute them into the formula:
z = (11,000 - 10,000) / 1,000
z = 1
Step 2: Find the area under the standard normal curve.
Using the z-score we calculated, we can find the corresponding area under the standard normal curve. This area represents the probability of a randomly selected lamp lasting 11,000 hours or longer.
Using a standard normal distribution table or a statistical calculator, we can find that the area to the left of z = 1 is approximately 0.8413.
Step 3: Compute the expected number of lamps.
To calculate the expected number of lamps, we need to multiply the probability we found in Step 2 by the total number of lamps in the order (8,000 in this case).
Expected number of lamps = probability * total number of lamps
Expected number of lamps = 0.8413 * 8,000
Expected number of lamps ≈ 6,730
Therefore, we can expect approximately 6,730 lamps to last 11,000 hours or longer in an order of 8,000 lamps.