Question

Airlines typically accept more reservations for a flight than the number of seats on the plane. Suppose that for a certain route, an airline accepts $40$reservations on a plane that carries $38$passengers. Based on experience, the probability distribution of $Y=$the number of passengers who actually show up for a randomly selected flight is given in the following table. You can check that ${\mu }_Y=37.4\mathrm{and}{\sigma }_Y=1.24.$

There is also a crew of two flight attendants and two pilots on each flight. Let $X=$the total number of people (passengers plus crew) on a randomly selected flight.

a. Make a graph of the probability distribution of $X$. Describe its shape.

b. Find and interpret role="math" ${\mu }_X$.

c. Calculate and interpret ${\sigma }_X$.

Short answer

a. The distribution is symmetric.

b. ${\mu }_X=41.4$

c.${\sigma }_X=1.24$

Step-by-step solution

Part(a) Step 1 : Given information

Given table :

Part(a) Step 2 : Simplification

Since there are $4$crews on each flight, so :

$X=Y+4$

So, the new table is :

And so, the respective graph is :

The distribution is symmetric.

Part(b) Step 1 : Given information

Given table :

Part(b) Step 2 : Simplification

$Y$stands for the number of passengers.

Each flight has four crew members, hence $X=Y+4$.

When the constant is added to each data value, the distribution's center is also enhanced by that constant value.

$$\begin{gathered} {\mu }_x={\mu }_y+4 \\ =37.4+5 \\ =\mathbf{41}\mathbf{.}\mathbf{4} \end{gathered}$$

When the constant value is added to each data point, the center of the distribution is raised, resulting in $41.4$average numbers of persons (passengers + staff) being randomly selected.

Part(c) Step 1 : Given information

Given table :

Part(c) Step 2 : Simplification

$Y$stands for the number of passengers.

Because each flight has four crew members, adding the constant to each data value has no effect on the distribution's spread; it remains intact.

$1.24={\sigma }_x={\sigma }_y$

When a constant value is added to each data point, the spread of the distribution is unaffected, and the number of persons (passengers + staff) for a randomly chosen aircraft varies on average $1.24$around the mean = $41.4$people.