Let's address each part of the problem step by step.

Given Data:

• Delivery Time (hours) $X$: 3, 4, 5, 6, 7
• Probability of each delivery time: $P(X = 3) = 0.20, \quad P(X = 4) = 0.35, \quad P(X = 5) = 0.25, \quad P(X = 6) = 0.15, \quad P(X = 7) = 0.05$

a. Mean (Expected Value) of the Delivery Time

The mean or expected value $E(X)$ is calculated using the formula: $E(X) = \sum [x_i \cdot P(x_i)]$ Where $x_i$ are the values of the random variable (delivery times) and $P(x_i)$ are the corresponding probabilities.

$E(X) = (3 \times 0.20) + (4 \times 0.35) + (5 \times 0.25) + (6 \times 0.15) + (7 \times 0.05)$

Let's calculate this value.

b. Variance of the Delivery Time

The variance $\text{Var}(X)$ is calculated using the formula: $\text{Var}(X) = \sum [(x_i - E(X))^2 \cdot P(x_i)]$ We first need to find the mean $E(X)$ (from part a) and then apply the formula.

c. Standard Deviation of the Delivery Time

The standard deviation $\sigma$ is simply the square root of the variance: $\sigma = \sqrt{\text{Var}(X)}$

d. Probability that Delivery Time is Less than or Equal to 5 Hours

We need to sum the probabilities where $X \leq 5$. This involves summing the probabilities for $X = 3$, $X = 4$, and $X = 5$.

$P(X \leq 5) = P(X = 3) + P(X = 4) + P(X = 5)$

e. Probability that Delivery Time is Greater than 6 Hours

This involves summing the probabilities for $X > 6$, i.e., for $X = 7$.

$P(X > 6) = P(X = 7)$

I'll now compute each of these values.### Results: a. Mean (Expected Value) of the Delivery Time: $E(X) = 4.5$ hours

b. Variance of the Delivery Time: $\text{Var}(X) = 1.25$

c. Standard Deviation of the Delivery Time: $\sigma = 1.12$ hours (approximately)

d. Probability that Delivery Time is Less than or Equal to 5 Hours: $P(X \leq 5) = 0.80$

e. Probability that Delivery Time is Greater than 6 Hours: $P(X > 6) = 0.05$

Would you like further details on any of these calculations, or have any questions?

Here are 5 related questions for practice:

1. What is the probability that the delivery time is exactly 6 hours?
2. What is the probability that the delivery time is greater than 4 hours?
3. How would the mean change if the delivery time of 3 hours had a higher probability?
4. What is the cumulative probability for delivery times less than 7 hours?
5. What is the z-score for a delivery time of 7 hours?

Tip: Variance measures the spread of the distribution, while the standard deviation provides this spread in the same units as the original data.