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A cement truck delivers mixed cement to a large construction site.
Let x represent the cycle time in minutes for the truck to leave the construction site,
go back to the cement plant, fill up, and return to the construction site
with another load of cement. From past experience, it is known that the mean cycle time
is 𝜇 = 48 minutes with 𝜎 = 18 minutes. The x distribution is approximately normal.
(a) What is the probability that the cycle time will exceed 60 minutes,
given that it has exceeded 50 minutes? (Round your answer to four decimal places.)
(b) What is the probability that the cycle time will exceed 55 minutes,
given that it has exceeded 40 minutes? (Round your answer to four decimal places.)
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In this my post, I will solve and answer question (a), ONLY.
I will leave question (b) to you, because it is a TWIN to question (a)
and, therefore, I expect that it will be easy to you to solve it on your own,
having a TEMPLATE in front of you.
So, trace my solution for question (a) very attentively.
This question is about conditional probability.
We have an event A = { x > 60 } and an event B = { x > 50 }.
They want you calculate the conditional probability P(A | B), which reads " P of A given B".
This conditional probability is the ratio .
Here P(A and B) is the probability of the intersection (A and B).
In this problem, OBVIOUSLY, event A is a part of the event B: indeed, if x > 60, then x > 50.
Therefore, P(A and B) = P(A) <<<---=== exactly because A is a smaller part of B.
So, our conditional probability P(A | B) is the ratio P(A | B) = .
+---------------------------------------------------------------------+
| As soon as we got this understanding, we solved more than half |
| of the problem, and the rest is just a technique. |
+---------------------------------------------------------------------+
Mow, P(A) = normalcdf(60, 9999, 48, 18) (it is the area under the normal curve
on the right of the raw mark x=60),
so, using your calculator, you get the value P(A) = 0.2525.
Next, P(B) = normalcdf(50, 9999, 48, 18) (it is the area under the normal curve
on the right of the raw mark x=50),
so, using your calculator, you get the value P(B) = 0.4558.
Finally, the desired conditional probability is
P(A | B) = = = 0.5540 (rounded). ANSWER
Thus the part (a) of the problem is just solved.
Now boldly go forward and solve part (b) in the same way.
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If you need to learn about using calculators to work with function normalcdf,
you can find all necessary instructions from this video-lesson
Normal Distribution: Calculating Probabilities {TI 84 Plus CE}
https://www.youtube.com/watch?v=IyEKEL9nm28
or from this textual description
https://cosmosweb.champlain.edu/people/stevens/WebTech/TIFiles/Chap6-TI-83.pdf
Happy learning, solving and calculating ( ! )