the scheduled commuting time on Long Island Rail Road from Glen Cove to New York is 65 minutes. Suppose that the actual commuting time is uniformly distributed between 64 and 74 minutes. What is the probability that the commuting time will be a) less than 70 minutes B) between 65 and 70 minutes c) greated than 65 minutes D) what are the mean and standard deviation of the commuting time
This problem has been posted and responded to previously.
If the times are "uniformly distributed," this means that the frequencies for all times are equal. Do you mean normally distributed?
no uniformly distribution
The scheduled commuting time on the Long Island Railroad from Glen Cove to New York City is 65 minutes. Suppose that the actual commuting time is uniformly distributed between 64 and 74 minutes. What is the probability that the commuting time will be
a. less than 70 minutes?
b. between 65 and 70 minutes?
c. greater than 65 minutes?
d. What are the means and standard deviation of the commuting time?
To find the probabilities of the commuting time, we can use the properties of a uniform distribution. In this case, we have a uniform distribution between 64 and 74 minutes.
a) To find the probability that the commuting time is less than 70 minutes, we need to calculate the area under the probability density function (PDF) curve from 64 to 70 minutes. Since the distribution is uniform, the PDF is a rectangle with a height of 1/10 (1 divided by the range of 10 minutes) and a width of 6 minutes (70 - 64).
The probability is equal to the area of the rectangle, which is given by the formula A = height × width:
A = (1/10) × 6 = 6/10 = 0.6
Therefore, the probability that the commuting time is less than 70 minutes is 0.6 or 60%.
b) To find the probability that the commuting time is between 65 and 70 minutes, we need to calculate the area under the PDF curve from 65 to 70 minutes. Again, the PDF is a rectangle with a height of 1/10 and a width of 5 minutes (70 - 65).
The probability is equal to the area of the rectangle:
A = (1/10) × 5 = 5/10 = 0.5
Therefore, the probability that the commuting time is between 65 and 70 minutes is 0.5 or 50%.
c) To find the probability that the commuting time is greater than 65 minutes, we need to calculate the area under the PDF curve from 65 to 74 minutes. This is equal to 1 minus the probability of the commuting time being less than or equal to 65 minutes.
The probability of being less than or equal to 65 minutes is calculated using the formula from part a:
A = (1/10) × (65 - 64) = 1/10 = 0.1
Therefore, the probability that the commuting time is greater than 65 minutes is 1 - 0.1 = 0.9 or 90%.
d) The mean of a uniform distribution is the average of the lower and upper limits. In this case, the mean is (64 + 74) / 2 = 68.
The standard deviation of a uniform distribution is given by the formula (upper limit - lower limit) / sqrt(12). In this case:
Standard deviation = (74 - 64) / sqrt(12) ≈ 2.58199 (rounded to 5 decimal places).
Therefore, the mean of the commuting time is 68 minutes, and the standard deviation is approximately 2.58199 minutes.