The taxi and takeoff time for commercial jets is a random variable x with a mean of 8.3 minutes and a standard deviation of 3.5 minutes. Assume that the distribution of taxi and takeoff times is approximately normal. You may assume that the jets are lined up on a runway so that one taxies and takes off immediately after the other, and that they take off one at a time on a given runway.
A button hyperlink to the SALT program that reads: Use SALT.
(a) What is the probability that for 35 jets on a given runway, total taxi and takeoff time will be less than 320 minutes? (Round your answer to four decimal places.)
(b) What is the probability that for 35 jets on a given runway, total taxi and takeoff time will be more than 275 minutes? (Round your answer to four decimal places.)
(c) What is the probability that for 35 jets on a given runway, total taxi and takeoff time will be between 275 and 320 minutes? (Round your answer to four decimal places.)
To solve these probability questions, we will use the properties of the normal distribution, specifically the properties of the z-score.
(a) To find the probability that the total taxi and takeoff time for 35 jets is less than 320 minutes, we need to standardize the value using the z-score formula:
z = (x - μ) / σ
where x is the given value, μ is the mean, and σ is the standard deviation.
In this case, x = 320, μ = 8.3, and σ = 3.5.
Using the z-score formula:
z = (320 - 8.3) / 3.5 ≈ 90.57
Next, we need to find the probability corresponding to this z-score using a standard normal distribution table or a statistical calculator. Let's use the SALT program mentioned:
[Use SALT]
Entering the z-score of 90.57 into the SALT program will give us the probability.
(b) To find the probability that the total taxi and takeoff time for 35 jets is more than 275 minutes, we need to again standardize the value using the z-score formula:
z = (x - μ) / σ
where x = 275, μ = 8.3, and σ = 3.5.
Using the z-score formula:
z = (275 - 8.3) / 3.5 ≈ 81.57
We can once again use the SALT program to find the probability corresponding to this z-score.
(c) To find the probability that the total taxi and takeoff time for 35 jets is between 275 and 320 minutes, we need to find the probability of both values separately and then subtract one from the other.
Using the z-score formula, we find the z-scores for 275 and 320 as mentioned earlier.
Let's denote the probability of being less than 275 as P1 and the probability of being less than 320 as P2.
Then, P1 = probability of total taxi and takeoff time less than 275 minutes
P2 = probability of total taxi and takeoff time less than 320 minutes
The required probability is:
P2 - P1
Using the SALT program mentioned earlier, we can find the values of P1 and P2 and calculate the final probability.
I hope this explanation helps you understand how to solve these probability questions using the properties of the normal distribution and the z-score formula.
To solve these probability questions, we need to use the properties of the normal distribution. Let's calculate each part step-by-step.
Step 1: Standardize the values
First, we need to standardize the given values using the Z-score formula:
Z = (X - μ) / σ
where:
X = the given value
μ = the mean
σ = the standard deviation
(a) For the total taxi and takeoff time of less than 320 minutes:
Z = (320 - 8.3 * 35) / (3.5 * √35)
(b) For the total taxi and takeoff time of more than 275 minutes:
Z = (275 - 8.3 * 35) / (3.5 * √35)
(c) For the total taxi and takeoff time between 275 and 320 minutes, we need to calculate the probabilities for both values and subtract the smaller probability from the larger one.
Z1 = (275 - 8.3 * 35) / (3.5 * √35)
Z2 = (320 - 8.3 * 35) / (3.5 * √35)
Step 2: Calculate the probabilities
(a) To find the probability of a Z-score less than a certain value, we can use a standard normal distribution table or a calculator. Let's assume you will use the SALT program.
P(Z < Z-value) = Use SALT
(b) To find the probability of a Z-score greater than a certain value, we can use the complementary probability:
P(Z > Z-value) = 1 - P(Z < Z-value)
(c) To find the probability between two Z-scores, we can subtract the smaller probability from the larger one:
P(Z1 < Z < Z2) = P(Z < Z2) - P(Z < Z1)
Now let's put all the steps together and solve the problem.