1. **Problem:** Find the values at given distances from the mean for a normal distribution with mean $\mu=45$ and standard deviation $\sigma=8.3$. 2. **Formula:** Value at $k$ standard deviations from the mean is given by: $$ x = \mu + k \times \sigma $$ 3. **Calculations:** - For $+1$ standard deviation: $$ x = 45 + 1 \times 8.3 = 45 + 8.3 = 53.3 $$ - For $+3$ standard deviations: $$ x = 45 + 3 \times 8.3 = 45 + 24.9 = 69.9 $$ - For $-1$ standard deviation: $$ x = 45 - 1 \times 8.3 = 45 - 8.3 = 36.7 $$ - For $-2$ standard deviations: $$ x = 45 - 2 \times 8.3 = 45 - 16.6 = 28.4 $$ 4. **Explanation:** Each value is found by moving the specified number of standard deviations away from the mean. Positive means to the right, negative to the left on the number line. 5. **Sketching the normal curve:** - The curve is bell-shaped centered at $\mu=45$. - Mark the x-axis at $45$ (mean), $53.3$ (+1 SD), $69.9$ (+3 SD), $36.7$ (-1 SD), and $28.4$ (-2 SD). - The curve is symmetric about the mean. Final answers: - $+1$ SD: $53.3$ - $+3$ SD: $69.9$ - $-1$ SD: $36.7$ - $-2$ SD: $28.4$