Question

Can you explain where we got 1.28 from?? and how to calculate

the standard deviation
7. A city installs 2000 electric lamps for street lighting. These lamps have a mean burning life of 1000 hours with a standard deviation of 200 hours. The normal distribution is a close approximation to this case. (a) What is the probability that a lamp will fail in the first 700 burning hours? (b) What is the probability that a lamp will fail between 900 and 1300 burning hours? (c) After how many burning hours would we expect 10% of the lamps to be left? Let X denote the random variable of the burning life of an electric lamp. X∼N(1000,200)
(a) P(X≤700)=FX(700)=ϕ(200700−1000)=ϕ(−1.5)=1−ϕ(1.5)=1−0.9332=0.0667
(b)P(900≤X≤1300)=FX(1300)−FX(900)=ϕ(2001300−1000)−ϕ(200900−1000)=ϕ(1.5)−ϕ(−0.5)=ϕ(1.5)−1+ϕ(0.5)=0.9332−1+0.6915=0.6247
(c) P(X>x)=0.1,P(X≤x)=0.9,ϕ(1.28)=0.9,200x−1000=1.28,x=1256

Answer 1

The number 1.28 is a z-score corresponding to the 90th percentile of a standard normal distribution. The standard deviation and mean are used to calculate the z-scores, which then help determine probabilities related to the burning life of electric lamps.

Step-by-step explanation:

The number 1.28 refers to a z-score in the context of a standard normal distribution. The z-score is a statistic that represents the number of standard deviations an element is from the mean of a distribution. In this particular case, the z-score of 1.28 corresponds to the 90th percentile, indicating that 90% of the data falls below this z-score value. The calculation of probabilities and the determination of specific hours where a certain percentage of lamps are left burning involves using the properties of the normal distribution.

Calculating the standard deviation is not directly shown in the question, as it is given as part of the problem's data. To use this information for probability calculations, you convert the values to z-scores by subtracting the mean from the observation and dividing by the standard deviation. These z-scores are then associated with probabilities using the standard normal distribution table or a calculator with the normal distribution function.