A large data set is approximately normally distributed. Three values in the data set are represented by a, b, and c where a is the value of the third quartile, b is the value with a -score of 1, and c is the value of the 70th percentile. List a, b, and c

To order the values a, b, and c—where: - \( a \) is the third quartile (Q3, which corresponds to the 75th percentile), - \( b \) is the value with a z-score of 1, - \( c \) is the value at the 70th percentile— you can use the following relationships based on the properties of a normal distribution: 1. **Third Quartile (Q3)**: The third quartile corresponds to the 75th percentile. This means Q3 is greater than 70% of the data. 2. **Z-score of 1**: A z-score of 1 indicates that this value is one standard deviation above the mean. In a standard normal distribution, about 84.13% of data points fall below this z-score, meaning the z-score of 1 is greater than the 70th percentile (which corresponds to a z-score of approximately 0.524). 3. **70th Percentile**: This value, denoted as \( c \), is less than Q3 (the 75th percentile) but greater than the mean. Based on this information, we can rank the values: - The 70th percentile (\( c \)) is the smallest. - The z-score of 1 (\( b \)) comes next, as it represents a value higher than the 70th percentile. - The third quartile (\( a \)) is the largest, as it signifies the 75th percentile. Thus, we can list them in order: **c < b < a**. Update (2024-12-27): To identify the values of \( a \), \( b \), and \( c \) in a normally distributed data set, we need to understand what these terms represent: 1. **\( a \)**: This is the value of the third quartile (Q3). In a normal distribution, Q3 corresponds to the 75th percentile. Given that the mean is represented by \( \mu \) and the standard deviation by \( \sigma \), we can calculate Q3 using the Z-score for the 75th percentile, which is approximately 0.674. Therefore, the value \( a \) can be calculated as: \[ a = \mu + 0.674 \sigma \] 2. **\( b \)**: This is the value with a Z-score of 1. The Z-score of a data point is calculated as: \[ Z = \frac{X - \mu}{\sigma} \] For a Z-score of 1, we can rearrange this equation to find the value: \[ b = \mu + 1\sigma = \mu + \sigma \] 3. **\( c \)**: This is the value of the 70th percentile. The Z-score for the 70th percentile is approximately 0.524. Thus, we can calculate \( c \) as: \[ c = \mu + 0.524 \sigma \] In summary, the expressions for \( a \), \( b \), and \( c \) in terms of the mean (\( \mu \)) and standard deviation (\( \sigma \)) are as follows: - \( a = \mu + 0.674 \sigma \) (third quartile) - \( b = \mu + \sigma \) (Z-score of 1) - \( c = \mu + 0.524 \sigma \) (70th percentile) These equations describe how \( a \), \( b \), and \( c \) are determined based on the properties of a normal distribution. To obtain specific numerical values, you'll need the values of \( \mu \) and \( \sigma \) from your data set.