Final answer:
The probability that the lifetime of a modem is longer than 780 hours, given a normal distribution with a mean of 1,000 hours and a standard deviation of 100 hours, can be found using the Z-score calculation and is approximately 0.9861 or 98.61%.
Step-by-step explanation:
To find the probability that the lifetime of a modem is longer than 780 hours, given that it has a normal distribution with a mean (μ) of 1,000 hours and a standard deviation (σ) of 100 hours, we will use the Z-score formula. The Z-score formula is Z = (X - μ) / σ, where X is the value we are examining. After calculating the Z-score, we will look up the corresponding probability in the standard normal distribution table or use a calculator with a normalcdf function.
First, calculate the Z-score for 780 hours:
Z = (780 - 1000) / 100
Z = -220 / 100
Z = -2.2
Now we look up or calculate the probability of Z being greater than -2.2. This is the same as looking for the probability that the modem lasts longer than 780 hours.
Using the standard normal distribution table or a calculator's normalcdf function, we find the area to the right of Z = -2.2, which represents the probability we're looking for. The probability of a modem lasting longer than 780 hours is approximately 0.9861 or 98.61%.
Therefore, there is a high probability that the lifetime of a modem will exceed 780 hours.