Answer:
The probability that the technician will finish a particular diagnostic job within 1.5 hours is 0.1667 or 16.67% (rounded to 2 decimal places).
Explanation:
We can use the cumulative distribution function (cdf) to find the probability that the technician will finish a particular diagnostic job within 1.5 hours.
Let X represent the time it takes the technician to complete the diagnostic job, where X is a continuous random variable between 1 and 4 hours. The cdf of X is given by:
F(x) = P(X ≤ x)
To find P(X ≤ 1.5), we evaluate the cdf at x = 1.5:
F(1.5) = P(X ≤ 1.5)
We can find F(x) by breaking up the range of X into two intervals:
For X ≤ 1, P(X ≤ x) = 0, since the technician needs at least 1 hour to complete the job.
For 1 < X ≤ 4, P(X ≤ x) = (x-1)/3, since the technician finishes the job in x hours with equal probability in this interval.
Therefore, we can write the cdf as:
F(x) = { 0 if x ≤ 1
{ (x-1)/3 if 1 < x ≤ 4
To find P(X ≤ 1.5), we substitute x = 1.5 into the cdf:
F(1.5) = (1.5-1)/3 = 0.1667
So the probability that the technician will finish the particular diagnostic job within 1.5 hours is 0.1667 or 16.67% (rounded to 2 decimal places).
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