To determine the range within which 68% of the students' scores fall, we need to consider the properties of a normal distribution.
1. Mean (Average Score): The mean score of the students is 57.5.
2. Standard Deviation: The standard deviation, which measures the dispersion or spread of the scores, is 6.5.
For a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. This means we need to calculate the values that are one standard deviation below the mean and one standard deviation above the mean.
1. Lower Bound: To find the lower bound, subtract one standard deviation from the mean.
[tex]\[ \text{Lower Bound} = \text{Mean} - \text{Standard Deviation} = 57.5 - 6.5 = 51.0 \][/tex]
2. Upper Bound: To find the upper bound, add one standard deviation to the mean.
[tex]\[ \text{Upper Bound} = \text{Mean} + \text{Standard Deviation} = 57.5 + 6.5 = 64.0 \][/tex]
Therefore, 68% of the students received scores between 51.0 and 64.0.
So, the correct answer is:
The scores of eighth-grade students in a math test are normally distributed with a mean of 57.5 and a standard deviation of 6.5. From this data, we can conclude that 68% of the students received scores between 51.0 and 64.0.
