To solve the problem, let’s break it down step by step:

Problem Context:

We are given:

• Two independent random variables $X$ and $Y$
• $\mu_X = 9.70$, $\sigma_X = 0.400$
• $\mu_Y = 6.8$, $\sigma_Y = 0.01$

We need to calculate the mean and standard deviation of $3X$, using these rules:

1. Mean of a scaled random variable: $\mu_{aX} = a \cdot \mu_X$
2. Standard deviation of a scaled random variable: $\sigma_{aX} = |a| \cdot \sigma_X$

Solution:

Step 1: Mean of $3X$

$\mu_{3X} = 3 \cdot \mu_X = 3 \cdot 9.70 = 29.10$

Step 2: Standard deviation of $3X$

$\sigma_{3X} = |3| \cdot \sigma_X = 3 \cdot 0.400 = 1.200$

Final Answer:

• Mean: $29.100$ (rounded to three decimal places)
• Standard Deviation: $1.200$ (rounded to three decimal places)

Do you want further explanations or help solving related problems? Here are 5 related questions:

1. How would you calculate the mean and standard deviation for $X + Y$?
2. How does scaling a random variable affect its variance instead of standard deviation?
3. What happens if $X$ and $Y$ are not independent—how would the variance change?
4. Could you demonstrate how the rules for linear transformations apply to $3X + 2Y$?
5. Can we interpret these results for a real-world example where $X$ is a physical measurement?

Tip: Always remember that variance scales with the square of the coefficient, while the standard deviation scales linearly.