Questions: Suppose the time it takes a barber to complete a haircuts is uniformly distributed between 5 and 17 minutes, inclusive. Let X= the time, in minutes, it takes a barber to complete a haircut. Then X ~ U(5,17). Find the probability that a randomly selected barber needs at least seven minutes to complete the haircut, P(x>7) (round to 4 decimal places) Answer: A

Solution Steps

The mean \( E(X) \) of a uniform distribution \( U(a, b) \) is calculated using the formula:

\[ E(X) = \frac{a + b}{2} \]

Substituting \( a = 5 \) and \( b = 17 \):

\[ E(X) = \frac{5 + 17}{2} = 11.0 \]

The variance \( \text{Var}(X) \) of a uniform distribution is given by:

\[ \text{Var}(X) = \frac{(b - a)^2}{12} \]

Substituting the values:

\[ \text{Var}(X) = \frac{(17 - 5)^2}{12} = \frac{144}{12} = 12.0 \]

The standard deviation \( \sigma(X) \) is the square root of the variance:

\[ \sigma(X) = \sqrt{\text{Var}(X)} = \sqrt{12.0} \approx 3.4641 \]

The cumulative distribution function \( F(x; a, b) \) for a uniform distribution is defined as:

\[ F(x; a, b) = \frac{x - a}{b - a}, \quad a \leq x \leq b \]

To find the probability that a randomly selected barber needs at least seven minutes to complete the haircut, we calculate:

\[ P(X \geq 7) = P(7 \leq X \leq 17) = F(17) - F(7) \]

Calculating \( F(17) \) and \( F(7) \):

\[ F(17) = \frac{17 - 5}{17 - 5} = 1.0 \] \[ F(7) = \frac{7 - 5}{17 - 5} = \frac{2}{12} = \frac{1}{6} \approx 0.1667 \]

Thus, the probability is:

\[ P(7 \leq X \leq 17) = 1.0 - 0.1667 = 0.8333 \]

Final Answer