For a certain type of light, the number of hours a bulb will burn before requiring replacement has a mean of 5000 hours and a standard deviation of 200 hours. Suppose that 7000 such bulbs are installed in an office building. Estimate the number that will require replacement between 4600 and 5400 hours from the time of installation.
Answer
Final Answer: The estimated number of bulbs that will require replacement between 4600 and 5400 hours from the time of installation is approximately \(\boxed{6682}\).
Step 1 ：We are given a problem of normal distribution. The mean (\(\mu\)) is 5000 hours and the standard deviation (\(\sigma\)) is 200 hours. We need to estimate the number of bulbs that will require replacement between 4600 and 5400 hours from the time of installation.
Step 2 ：We start by calculating the z-scores for 4600 and 5400 hours. The z-score is calculated as \((X - \mu) / \sigma\). For 4600 hours, the z-score is -2.0 and for 5400 hours, the z-score is 2.0.
Step 3 ：Next, we find the area under the normal distribution curve between these two z-scores. This gives us the probability that a bulb will burn out between 4600 and 5400 hours. The probability is approximately 0.9545.
Step 4 ：Finally, we multiply this probability by the total number of bulbs (7000) to get the estimated number of bulbs that will burn out between 4600 and 5400 hours. The estimated number of bulbs is approximately 6682.
Step 5 ：Final Answer: The estimated number of bulbs that will require replacement between 4600 and 5400 hours from the time of installation is approximately \(\boxed{6682}\).
