Question

Let X represent the number of occupants in a randomly chosen car on a certain stretch of highway during morning commute hours. A survey of cars showed that the probability distribution of X is as follows.
X| 1 2 3 4 5
P(X)| 0.65 0.13 0.10 0.03 0.09

(g) To save energy, a goal is set to have the mean number of occupants to be at least two per car. Has this goal been met?

Since the computed mean number of occupants per car is
[blank], then the goal to have the mean number of occupants to be at least two per car [has or has not?] been met.

Answer 1

To determine whether the goal of having the mean number of occupants be at least two per car has been met, we'll first need to calculate the expected value (mean) of the random variable \( X \) using the probability distribution provided.

The mean \( E(X) \) can be calculated using the formula:

\[
E(X) = \sum (X \cdot P(X))
\]

Now we can substitute the values from the distribution:

\[
E(X) = (1 \cdot 0.65) + (2 \cdot 0.13) + (3 \cdot 0.10) + (4 \cdot 0.03) + (5 \cdot 0.09)
\]

Calculating each term:

1. \( 1 \cdot 0.65 = 0.65 \)
2. \( 2 \cdot 0.13 = 0.26 \)
3. \( 3 \cdot 0.10 = 0.30 \)
4. \( 4 \cdot 0.03 = 0.12 \)
5. \( 5 \cdot 0.09 = 0.45 \)

Now sum these values:

\[
E(X) = 0.65 + 0.26 + 0.30 + 0.12 + 0.45 = 1.78
\]

Now we have the mean number of occupants per car:

\[
E(X) = 1.78
\]

Now we can complete the statement:

Since the computed mean number of occupants per car is **1.78**, then the goal to have the mean number of occupants to be at least two per car **has not** been met.