12. Let N be a random variable defined as the number of hours slept per night by college seniors. Let’s suppose no one sleeps less than four hours or more than eight hours per night. The following probability function is given for N.

Question

f(N) = {0.15 for n = 4, 0.23 for n = 5, 0.19 for n = 6, 0.24 for n = 7, and 0.19 for n = 8}
What is the expected value for N? Give your answer as a decimal.

Answer 1

To find the expected value \( E(N) \) of the random variable \( N \), we can use the formula for the expected value of a discrete random variable:

\[ E(N) = \sum_{n} n \cdot f(n) \]

where \( n \) are the possible outcomes and \( f(n) \) is the probability of each outcome.

Given the probability function:

• \( f(4) = 0.15 \)
• \( f(5) = 0.23 \)
• \( f(6) = 0.19 \)
• \( f(7) = 0.24 \)
• \( f(8) = 0.19 \)

We can now calculate the expected value:

\[ E(N) = 4 \cdot f(4) + 5 \cdot f(5) + 6 \cdot f(6) + 7 \cdot f(7) + 8 \cdot f(8) \]

Substituting in the values:

\[ E(N) = 4 \cdot 0.15 + 5 \cdot 0.23 + 6 \cdot 0.19 + 7 \cdot 0.24 + 8 \cdot 0.19 \]

Now calculating each term:

• \( 4 \cdot 0.15 = 0.60 \)
• \( 5 \cdot 0.23 = 1.15 \)
• \( 6 \cdot 0.19 = 1.14 \)
• \( 7 \cdot 0.24 = 1.68 \)
• \( 8 \cdot 0.19 = 1.52 \)

Now, summing these values:

\[ E(N) = 0.60 + 1.15 + 1.14 + 1.68 + 1.52 \] \[ E(N) = 5.09 \]

Thus, the expected value for \( N \) is:

\[ \boxed{5.09} \]