The school board is considering reallocating $1,000 per student. If this funding were given to 10 students, what would be the predicted change in the graduation rate?

The regression equation is $\hat{Y} = 82 + 0.04X$.
X represents school funding in thousands of dollars per student.
The school board proposes reallocating $1,000 per student. The question asks for the change if funding were given to 10 students.

Total funding change = 1,000 \text{ per student} \times 10 \text{ students} = \10,000$. Since X is in thousands of dollars,$\Delta X = \frac{$10,000}{$1,000/\text{thousand}} = 10$.

The change in $\hat{Y}$ is given by the slope multiplied by the change in X.

The predicted graduation rate change is $\boxed{0.4 \text{ percentage points}}$.

2b.iii)
To determine homogeneity, calculate the Coefficient of Variation (CV) for each school. A smaller CV indicates more homogeneous performance.
The formula for CV is $\text{CV} = \frac{\text{Standard Deviation}}{\text{Mean}} \times 100\%$.

Mean = 75, SD = 5

Mean = 70, SD = 12

Since $\text{CV}_A (6.67\%)$ is less than $\text{CV}_B (17.14\%)$, School A has more homogeneous performance.
School A has the most homogeneous performance because its coefficient of variation is lower, indicating that its students' scores are less spread out relative to their mean score compared to School B.

3a.i)

For the top 10%, we need the score $X$ such that $P(Z > z) = 0.10$, which means $P(Z \le z) = 0.90$.
Using a standard normal distribution table or calculator, the Z-score for the 90th percentile is approximately $z = 1.28$.

Given $\mu = 75$ and $\sigma = 8$.

The score that corresponds to an A- (top 10%) is $\boxed{85.24}$.

3a.ii)

Failing is defined as the bottom 5%.

Number of failing students = $100 \text{ students} \times 0.05 = 5 \text{ students}$.
In a 100-person class, $\boxed{5 \text{ students}}$ would theoretically fail