1. **State the problem:** We want to understand the formula for the mean maximum value $b$ given by $$b = a + \frac{3s}{\sqrt{n}}$$ where $a$ is the school mean, $s$ is the sample spread about the mean, and $n$ is the sample size. 2. **Explain the formula:** - The term $a$ represents the baseline mean value. - The term $\frac{3s}{\sqrt{n}}$ is an adjustment that depends on the sample spread and size. - As $n$ increases, $\sqrt{n}$ increases, making $\frac{3s}{\sqrt{n}}$ smaller, so $b$ approaches $a$. 3. **Important rules:** - The square root function $\sqrt{n}$ grows slower than $n$. - The fraction $\frac{3s}{\sqrt{n}}$ decreases as $n$ increases. 4. **Intermediate work:** - For example, if $a=50$, $s=10$, and $n=25$, then $$b = 50 + \frac{3 \times 10}{\sqrt{25}} = 50 + \frac{30}{5} = 50 + 6 = 56$$ 5. **Interpretation:** - The graph shows $b$ starting at $a$ when $n$ is small and increasing as $n$ grows, but the increase slows down because of the $\frac{1}{\sqrt{n}}$ term. - This means the maximum mean value $b$ is always above $a$ by an amount that decreases as the sample size $n$ increases. **Final answer:** The formula $b = a + \frac{3s}{\sqrt{n}}$ models how the mean maximum value $b$ depends on the school mean $a$, sample spread $s$, and sample size $n$, with $b$ approaching $a$ as $n$ becomes large.