History Grades In a history class, the grades for various assessments are all positive numbers and have different distributions. Determine whether the grades for each assessment could be normally distributed. Explain your reasoning.


b. a final with a mean of 72, standard deviation of 9, and 90th percentile score of 93

Verified step by step guidance

1. Step 1: Understand the characteristics of a normal distribution. A normal distribution is symmetric, bell-shaped, and defined by its mean (μ) and standard deviation (σ). It also follows the empirical rule, where approximately 68% of the data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.
2. Step 2: Use the given information to verify if the data aligns with a normal distribution. The problem provides a mean (μ) of 72, a standard deviation (σ) of 9, and a 90th percentile score of 93. In a normal distribution, the 90th percentile corresponds to a z-score of approximately 1.28.
3. Step 3: Calculate the theoretical value of the 90th percentile using the z-score formula: z = (X - μ) / σ. Rearrange the formula to solve for X (the score): X = μ + z * σ. Substitute the given values: μ = 72, σ = 9, and z = 1.28.
4. Step 4: Compare the calculated 90th percentile score to the given value of 93. If the calculated value closely matches 93, it supports the assumption that the data could be normally distributed. If there is a significant discrepancy, it may indicate that the data is not normally distributed.
5. Step 5: Consider other factors that could affect normality, such as the shape of the data distribution, outliers, or skewness. While the calculations provide a numerical check, visualizing the data (e.g., using a histogram or Q-Q plot) would offer additional evidence to confirm or refute normality.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. Understanding this concept is crucial for determining if a set of grades can be considered normally distributed.

Mean and Standard Deviation

The mean is the average of a set of values, while the standard deviation measures the amount of variation or dispersion in a set of values. In the context of the final exam grades, the mean of 72 indicates the central tendency, and the standard deviation of 9 indicates how spread out the grades are around the mean. These statistics help assess the distribution of grades.

Percentiles

A percentile is a measure used in statistics indicating the value below which a given percentage of observations fall. For example, the 90th percentile score of 93 means that 90% of the students scored below 93. This concept is important for understanding the distribution of grades and assessing whether the data aligns with the characteristics of a normal distribution.