1. **Problem Statement:** The scores of Senior High School students in their Statistics and Probability exam are normally distributed with mean $\mu=35$ and standard deviation $\sigma=5$. We need to answer several questions about this distribution. 2. **Formula and Rules:** For a normal distribution, the z-score formula is: $$z=\frac{x-\mu}{\sigma}$$ where $x$ is a score, $\mu$ is the mean, and $\sigma$ is the standard deviation. The Empirical Rule states: - About 68% of data falls within $\mu \pm 1\sigma$ - About 95% falls within $\mu \pm 2\sigma$ - About 99.7% falls within $\mu \pm 3\sigma$ 3. **Part a: Percent of scores between 30 and 40** Calculate z-scores: $$z_{30}=\frac{30-35}{5} = \frac{-5}{5} = -1$$ $$z_{40}=\frac{40-35}{5} = \frac{5}{5} = 1$$ Using the Empirical Rule, about 68% of scores lie between $z=-1$ and $z=1$, so approximately 68% of scores are between 30 and 40. 4. **Part b: Scores within 95% of the distribution** 95% corresponds to $\mu \pm 2\sigma$: $$35 - 2(5) = 35 - 10 = 25$$ $$35 + 2(5) = 35 + 10 = 45$$ So, scores between 25 and 45 fall within 95% of the distribution. 5. **Part c: Scores below and above the mean** Since the distribution is symmetric about the mean: - 50% of scores fall below the mean (less than 35) - 50% of scores fall above the mean (greater than 35) 6. **Part d: Constructing the normal curve and using the Empirical Rule** The normal curve is centered at 35 with spread determined by $\sigma=5$. The Empirical Rule intervals are: - 68% between 30 and 40 - 95% between 25 and 45 - 99.7% between 20 and 50 These intervals help us understand the distribution of scores visually and probabilistically.