Questions: A daily mail is delivered to your house between 1:00 p.m. and 5:00 p.m. Assume delivery times follow the continuous uniform distribution. Determine the percentage of mail deliveries that are made after 4:00 p.m. Multiple Choice 37.5% 33.3% 25% 27.5%

A daily mail is delivered to your house between 1:00 p.m. and 5:00 p.m. Assume delivery times follow the continuous uniform distribution. Determine the percentage of mail deliveries that are made after 4:00 p.m. Multiple Choice 37.5% 33.3% 25% 27.5%
Transcript text: A daily mail is delivered to your house between 1:00 p.m. and 5:00 p.m. Assume delivery times follow the continuous uniform distribution. Determine the percentage of mail deliveries that are made after 4:00 p.m. Multiple Choice 37.5\% 33.3\% 25\% 27.5\%
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Solution

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Solution Steps

Step 1: Define the Uniform Distribution

The delivery times of the mail are uniformly distributed between \( a = 13.0 \) (1:00 p.m.) and \( b = 17.0 \) (5:00 p.m.). The uniform distribution is characterized by its constant probability density function over the interval \([a, b]\).

Step 2: Calculate the Mean

The mean \( E(X) \) of a uniform distribution is given by the formula: \[ E(X) = \frac{a + b}{2} = \frac{13.0 + 17.0}{2} = 15.0 \]

Step 3: Calculate the Variance

The variance \( \text{Var}(X) \) of a uniform distribution is calculated using the formula: \[ \text{Var}(X) = \frac{(b - a)^2}{12} = \frac{(17.0 - 13.0)^2}{12} = \frac{16.0}{12} = 1.3333 \]

Step 4: Calculate the Standard Deviation

The standard deviation \( \sigma(X) \) is the square root of the variance: \[ \sigma(X) = \sqrt{\text{Var}(X)} = \sqrt{1.3333} \approx 1.1547 \]

Step 5: Calculate the Probability of Delivery After 4:00 p.m.

To find the probability of mail delivery after 4:00 p.m. (i.e., \( P(16.0 \leq X \leq 17.0) \)), we use the cumulative distribution function \( F(x; a, b) \): \[ F(x; a, b) = \frac{x - a}{b - a}, \quad a \leq x \leq b \] Calculating \( F(17.0) \) and \( F(16.0) \): \[ F(17.0) = \frac{17.0 - 13.0}{17.0 - 13.0} = 1.0 \] \[ F(16.0) = \frac{16.0 - 13.0}{17.0 - 13.0} = \frac{3.0}{4.0} = 0.75 \] Thus, the probability is: \[ P(16.0 \leq X \leq 17.0) = F(17.0) - F(16.0) = 1.0 - 0.75 = 0.25 \]

Final Answer

The percentage of mail deliveries that are made after 4:00 p.m. is \( 25\% \). Therefore, the answer is \[ \boxed{25\%} \]

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