(i) In the first 800 hours, approximately 1,587 lamps might be expected to fail.
(ii) Between 800 and 1200 hours, around 6,826 lamps might be expected to fail.
(iii) After 744 hours, 10% of the lamps would be expected to fail.
(iv) After 744 hours, around 90% of the lamps would be expected to still be burning.
This scenario involves using the normal distribution to estimate the number of lamps failing within certain time frames and determining when specific percentages of lamps fail or are still operational.
Given:
Mean (μ) = 1000 hours
Standard deviation (σ) = 200 hours
Total number of lamps = 10,000
We'll use the properties of the normal distribution to answer the questions:
(i) In the first 800 burning hours:
To find the probability of failure within 800 hours:
Z= X−μ/ σ
where X = 800 hours
Calculating Z:
Z= 800−1000/ 200 =−1
Using a standard normal distribution table or calculator, the probability of failure within 800 hours (Z = -1) is approximately 0.1587.
So, the expected number of lamps failing in the first 800 hours:
0.1587×10,000=1587
(ii) Between 800 and 1200 burning hours:
To find the probability of failure between 800 and 1200 hours, we'll find the probabilities for each and subtract:
For 800 hours:
Z_800 = 800−1000/ 200 =−1
Probability at 800 hours = 0.1587
For 1200 hours:
Z_1200 = 1200−1000/ 200 =1
Probability at 1200 hours = 0.8413
Probability between 800 and 1200 hours = Probability at 1200 hours - Probability at 800 hours
0.8413−0.1587=0.6826
Expected number of lamps failing between 800 and 1200 hours:
0.6826×10,000=6826
(iii) After how many burning hours would you expect 10% of the lamps to fail?
To find the time when 10% of lamps fail, we'll use the inverse of the cumulative distribution function (CDF) of the normal distribution.
Inverse CDF formula
X=μ+Z×σ
For 10% probability:
Find the Z-score corresponding to 10% in the standard normal distribution table, which is approximately -1.28.
Calculating the time:
X=1000+(−1.28×200)=744 hours
(iv) After how many burning hours would you expect 10% of the lamps to still be burning?
Since 10% have failed, 90% (100% - 10%) are still burning.
We need to find the time at which 90% of lamps are still burning. This is the inverse of the time when 10% have failed, so it's the same as the time found in (iii):
After 744 hours, approximately 10% of the lamps are expected to have failed, so about 90% of the lamps would still be burning.
Question
The local authorities in a certain city instal 10,000 electric lamps in the streets of the city. if these lamps have an average life of 1,000 burning hours with a standard deviation of 200 h, how many lamps might be expected to fail
(i) in the first 800 burning hours?
(ii) between 800 and 1200 burning hours? after how many burning hours would you expect
(iii) 10% of the lamps to fail?
(iv) 10% of the lamps to be still burning? assume that the life of lamps is normally distributed