The question asks us to calculate $E(Z)$ and the variance of $Z$ based on the transformation $Z = 5X + 3$. Here's how to solve it step-by-step:

Given Information:

1. $E(X) = 1.9$
2. $\text{Var}(X) = 0.89$
3. The transformation is $Z = 5X + 3$.

Step 1: Compute the mean of $Z$, $E(Z)$

Using the linear transformation rule for expectations: $E(Z) = 5E(X) + 3$ Substitute $E(X) = 1.9$: $E(Z) = 5(1.9) + 3 = 9.5 + 3 = 12.5$

So, $E(Z) = 12.5$.

Step 2: Compute the variance of $Z$, $\text{Var}(Z)$

Using the rule for variance under linear transformation: $\text{Var}(Z) = 5^2 \text{Var}(X)$ Substitute $\text{Var}(X) = 0.89$: $\text{Var}(Z) = 25(0.89) = 22.25$

Final Answers:

1. $E(Z) = 12.5$
2. $\text{Var}(Z) = 22.25$

Let me know if you'd like further clarification or a deeper explanation.

Follow-up Questions:

1. What are the rules for expectations and variances of linear transformations?
2. How does changing the coefficients of $Z = aX + b$ affect the variance and mean?
3. What happens to $\text{Var}(Z)$ if $b$ in $Z = aX + b$ is negative?
4. Can you explain why adding a constant to $X$ only affects the mean, not the variance?
5. How would the problem change if $X$ was transformed to $Z = 3X - 2$?

Tip:

Always remember: multiplying by a constant $a$ scales the variance by $a^2$, while adding a constant $b$ only shifts the mean without affecting the variance!