Question

For each probability and percentile problem, draw the picture. The speed of cars passing through the intersection of Blossom Hill Road and the Almaden Expressway varies from 14 to 35 mph and is uniformly distributed. None of the cars travel over 35 mph through the intersection. Part (a) In words, define the Random Variable X. - X: the number of cars passing through the intersection - X: the speed, in mph, of an individual car passing through the intersection - X: the number of cars driving below 35 mph through the intersection - X: the time, in seconds, it takes a car to pass through the intersection Part (b) Give the distribution of X. X ~ Uniform(14, 35), where 14 ≤ X ≤ 35 Part (e) Find the mean of X. μ = (14 + 35) / 2 = 24.5 Part (f) Find the standard deviation of X. (rounded to two decimal places) σ = (35 - 14) / √12 ≈ 6.12 Part (g) What is the probability that the speed of a car is at most 28 mph? P(X ≤ 28) = (28 - 14) / (35 - 14) = 14 / 21 = 2/3 Part (h) What is the probability that the speed of a car is between 18 and 24 mph? P(18 ≤ X ≤ 24) = (24 - 18) / (35 - 14) = 6 / 21 = 2/7 Part (i) State "P(22 < X < 52) = ___" in a probability question. What is the probability that the speed of a car is exactly 22 or 52 mph? P(22 < X < 52) = 0, since the speed of cars passing through the intersection is limited to a range of 14 to 35 mph. What is the probability that the speed of a car is below 22 or above 52 mph? P(X < 22 or X > 52) = 0, since the speed of cars passing through the intersection is limited to a range of 14 to 35 mph. What is the probability that the speed of a car is between 22 and 52 mph? P(22 ≤ X ≤ 52) = 1, since the speed of cars passing through the intersection is limited to a range of 14 to 35 mph. What is the probability that the speed of a car is below 22 given that it is below 52 mph? P(X < 22 | X < 52) = P(X < 22) = (22 - 14) / (35 - 14) = 8 / 21 Draw the picture and find the probability. (Enter your answer as a fraction.) Part (j) Find the 80th percentile. This means that 80% of the time, the speed is less than X mph while passing through the intersection. Part (k) Find the 85th percentile. In a complete sentence, state what this means. This means that 85% of the time, the speed is less than X mph while passing through the intersection. Part (l) Find the probability that the speed is more than 25 mph given (or knowing that) it is at least 17 mph. P(X > 25 | X ≥ 17) = (35 - 25) / (35 - 14) = 10 / 21

          For each probability and percentile problem, draw the picture.

The speed of cars passing through the intersection of Blossom Hill Road and the Almaden Expressway varies from 14 to 35 mph and is uniformly distributed. None of the cars travel over 35 mph through the intersection.

Part (a)
In words, define the Random Variable X.
- X: the number of cars passing through the intersection
- X: the speed, in mph, of an individual car passing through the intersection
- X: the number of cars driving below 35 mph through the intersection
- X: the time, in seconds, it takes a car to pass through the intersection

Part (b)
Give the distribution of X.
X ~ Uniform(14, 35), where 14 ≤ X ≤ 35

Part (e)
Find the mean of X.
μ = (14 + 35) / 2 = 24.5

Part (f)
Find the standard deviation of X. (rounded to two decimal places)
σ = (35 - 14) / √12 ≈ 6.12

Part (g)
What is the probability that the speed of a car is at most 28 mph?
P(X ≤ 28) = (28 - 14) / (35 - 14) = 14 / 21 = 2/3

Part (h)
What is the probability that the speed of a car is between 18 and 24 mph?
P(18 ≤ X ≤ 24) = (24 - 18) / (35 - 14) = 6 / 21 = 2/7

Part (i)
State "P(22 < X < 52) = ___" in a probability question.
What is the probability that the speed of a car is exactly 22 or 52 mph?
P(22 < X < 52) = 0, since the speed of cars passing through the intersection is limited to a range of 14 to 35 mph.

What is the probability that the speed of a car is below 22 or above 52 mph?
P(X < 22 or X > 52) = 0, since the speed of cars passing through the intersection is limited to a range of 14 to 35 mph.

What is the probability that the speed of a car is between 22 and 52 mph?
P(22 ≤ X ≤ 52) = 1, since the speed of cars passing through the intersection is limited to a range of 14 to 35 mph.

What is the probability that the speed of a car is below 22 given that it is below 52 mph?
P(X < 22 | X < 52) = P(X < 22) = (22 - 14) / (35 - 14) = 8 / 21

Draw the picture and find the probability. (Enter your answer as a fraction.)

Part (j)
Find the 80th percentile.
This means that 80% of the time, the speed is less than X mph while passing through the intersection.

Part (k)
Find the 85th percentile. In a complete sentence, state what this means.
This means that 85% of the time, the speed is less than X mph while passing through the intersection.

Part (l)
Find the probability that the speed is more than 25 mph given (or knowing that) it is at least 17 mph.
P(X > 25 | X ≥ 17) = (35 - 25) / (35 - 14) = 10 / 21

Added by Sandra O.

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