SOLUTION: An electrical company claims that the average life of the bulbs it manufactures is 1200 hours within a standard deviation of 250 hours. If a random sample of 100 bulbs is chosen, w
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Question 1174396: An electrical company claims that the average life of the bulbs it manufactures is 1200 hours within a standard deviation of 250 hours. If a random sample of 100 bulbs is chosen, what is the probability that the sample mean will be greater than 1150 hours Answer by CPhill(1987) (Show Source):
You can put this solution on YOUR website! **1. Calculate the Standard Error:**
* The standard error of the mean (SEM) measures the variability of the sample mean.
* SEM = σ / √n
* σ = population standard deviation (250 hours)
* n = sample size (100 bulbs)
* SEM = 250 / √100 = 250 / 10 = 25 hours
**2. Calculate the z-score:**
* The z-score tells us how many standard errors the sample mean is away from the population mean.
* z = (x̄ - μ) / SEM
* x̄ = sample mean (1150 hours)
* μ = population mean (1200 hours)
* SEM = standard error of the mean (25 hours)
* z = (1150 - 1200) / 25 = -50 / 25 = -2
**3. Find the Probability:**
* We want the probability that the sample mean is greater than 1150 hours. This is the same as finding the area to the right of z = -2 in the standard normal distribution.
* Using a z-table or calculator, look up the probability for z = -2. You'll find a value of approximately 0.0228.
* Since we want the area to the *right* of z = -2, subtract this value from 1: 1 - 0.0228 = 0.9772
**Therefore, the probability that the sample mean will be greater than 1150 hours is approximately 0.9772.**
