Question

Toby's Trucking Company determined that on an annual basis the distance traveled per truck is normally distributed with a mean of 50.0 thousand miles and a standard deviation of 12.0 thousand miles.
a. What proportion of trucks can be expected to travel between 34.0 and 50.0 thousand miles in the year?
b. What is the probability that a randomly selected truck travels between 34.0 and 38.0 thousand miles in the year?
c. What percentage of trucks can be expected to travel either below 30.0 or above 60.0 thousand miles in the year?
d. How many of the 1,000 trucks in the fleet are expected to travel between 30.0 and 60.0 thousand miles in the year?
e. How many miles will be traveled by at least 80% of the trucks?
f. What will your answers to be (a)-(e) if the standard deviation is 10.0 thousand miles?

Answer 1

Concepts and reason

The concept of normal distribution and standard normal variate is used to solve this problem.

A normal distribution is a probability distribution with the mean as and standard deviation as. The normal probability curve is a type of curve which is bell shaped.

The standard normal variate is a normal distribution which has mean equal to zero and standard deviation equal to one and the curve of this variate is bell shaped.

Fundamentals

Normal distribution is a probability distribution with the parameters and the continuous random variable X follows a normal distribution,

Where the mean isand the variance is. The formula to find the Z-Score is,

The probability of z is calculated by using the Excel function, .

The of a probability is calculated by using the Excel function, .

(a)

Consider the random variable X which represent the distance travelled by truck. The random variable is normally distributed with mean thousand miles and standard deviation thousand miles.

The probability that the expected distance travelled by truck is between thousand miles and thousand miles is calculated as:

The probabilities for the standard normal distribution are calculated by using Excel:

And

So,

(b)

The probability that truck travels between 34.0 and 38.0 thousand miles in the year can be calculated as:

The probabilities are calculated using Excel:

And,

So, the required probability is,

(c)

The percentage of trucks that are expected to travel less than 30 thousand miles can be calculated as:

The percentage of trucks that are expected to travel more than 60 thousand miles can be calculated as:

The probabilities are calculated using Excel:

And.

So, the required probability can be calculated as:

(d)

Use the concept of complementary event, to find the probability that a truck travels between 30 thousand miles and 60 thousand miles by subtracting the probability that a truck travels below 30 thousand miles or more than 60 thousand miles from 1 as:

So to find the trucks travel between 30 thousand miles and 60 thousand miles per thousand is calculated by multiplied the probability by 1000.

(e)

The expected distance in thousand miles travelled by at least of the trucks can be calculated as:

The Z-score corresponding to probability 0.8 can be computed using Excel as:

Hence, the calculation is,

(f.1)

Consider the random variable X which represent the distance travelled by truck. The mean is thousand miles and standard deviation isthousand miles.

The probability that the expected travel by truck is between thousand miles and thousand miles is calculated as,

The probabilities are calculated using Excel:

And,

So, the required probability can be calculated as:

(f.2)

The probability of the trucks that travel between 34.0 thousand miles to 38.0 thousand miles is calculated as,

The probabilities are calculated using Excel:

And,

So, the required probability can be calculated as:

(f.3)

The percentage of trucks that travel less than 30 thousand miles is calculated as:

The percentage of trucks that travel more than 60 thousand miles is calculated as:

The probabilities are calculated using Excel:

And,

So,

(f.4)

Use the concept of complementary event to find the probability that a track travel between 30 thousand miles and 60 thousand miles subtract the probability that a truck travels below 30 thousand miles or more than 60 thousand miles,

So to find the trucks travel between 30 thousand miles and 60 thousand miles per thousand is calculated by multiplied the probability by 1000.

(f.5)

The expected distance in thousand miles travelled by at least of the trucks can be calculated as:

The Z-score corresponding to probability 0.8 can be computed using Excel as:

Hence, the calculation is,

Ans: Part a

The proportion of the trucks that is expected to travel between 34 thousand to 50 thousand miles is .

Part b

The probability that the randomly selected truck travels between 34 thousand miles to 38 thousand miles is 0.066.

Part c

The percentage of trucks that are expected to travel below 30.0 or above 60.0 thousand miles is .

Part d

The number of trucks that are expected to travel between 30 thousand miles and 60 thousand miles is 750.

Part e

The distance in thousand miles travelled by at least of the trucks is 60.12.

Part f.1

The proportion of the trucks that travel between 34.0 thousand miles to 50.0 thousand miles is .

Part f.2

The probability that the randomly selected truck will travel between 34.0 thousand miles to 38.0 thousand miles is 0.06.

Part f.3

The percentage of trucks that are expected to travel below 30.0 thousand miles or above 60.0 thousand miles is .

Part f.4

The number of trucks that travel between 30 thousand miles and 60 thousand miles is 819.

Part f.5

The distance in thousand miles travelled by at least of the trucks is 58.42.