A telephone company's goal is to have no more than 4 monthly line failures on any 100 miles of line. The company currently experiences an average of 5 monthly line failures per 50 miles of line. Let x denote the number of monthly line failures per 100 miles of line. Assuming x has a Poisson distribution:
To find the distribution of the number of monthly line failures per 100 miles of line, x, we can use the Poisson distribution. The Poisson distribution is a probability distribution that describes the number of events that occur in a fixed interval of time or space.
In this case, the average number of monthly line failures per 50 miles of line is given as 5. Therefore, we can use this value to calculate the expected number of monthly line failures per 100 miles of line.
The formula for the Poisson distribution is:
P(x, μ) = (e^(-μ) * μ^x) / x!
Where P(x, μ) is the probability of x events occurring in the interval, μ is the average number of events in the interval, e is the base of natural logarithms (approximately 2.71828), and x! is the factorial of x.
In our case, x represents the number of monthly line failures per 100 miles of line. We want to find the probability that x is less than or equal to 4.
To solve this, we calculate the cumulative probability by summing up the probabilities for x = 0, 1, 2, 3, and 4. We can use a calculator or software that can evaluate the Poisson distribution, or we can use a Poisson table to find the probabilities and sum them up.
Once we have the cumulative probability, we can check if it is less than or equal to 0.05. If it is, then the telephone company's goal of having no more than 4 monthly line failures on any 100 miles of line is satisfied. If the cumulative probability is greater than 0.05, then the goal is not met.
By applying the Poisson distribution and calculating the cumulative probability, we can determine if the telephone company's goal is met.