1. **Problem statement:** We have a class of 150 students with exam grades normally distributed with mean $\mu = 67$ and standard deviation $\sigma = 8.5$. We want to find: a. The grade corresponding to a Z score of 6.7. b. The minimum grade to be in the 84th percentile. 2. **Formula and explanation:** The Z score formula is: $$Z = \frac{X - \mu}{\sigma}$$ where $X$ is the grade, $\mu$ is the mean, and $\sigma$ is the standard deviation. To find $X$ from $Z$, rearrange: $$X = Z \times \sigma + \mu$$ For percentiles, the Z score corresponding to the percentile can be found from standard normal distribution tables or using inverse normal functions. 3. **Part a: Find grade for Z = 6.7** $$X = 6.7 \times 8.5 + 67$$ Calculate: $$X = 56.95 + 67 = 123.95$$ Since grades typically max at 100, a Z score of 6.7 is extremely high and likely unrealistic, but mathematically the grade is approximately 123.95. 4. **Part b: Find minimum grade for 84th percentile** The Z score for the 84th percentile is approximately 1 (since 84th percentile corresponds to about 1 standard deviation above the mean). Calculate: $$X = 1 \times 8.5 + 67 = 75.5$$ So, the minimum grade to be in the 84th percentile is approximately 75.5. **Final answers:** a. Grade for Z=6.7 is approximately 123.95. b. Minimum grade for 84th percentile is approximately 75.5.