The National Assessment of Educational Progress (NAEP) is a nationwide assessment of students' proficiency in nine subjects: mathematics, reading, writing, science, the arts, civics, economics, geography, and U.S. history. The main NAEP assessments are conducted annually on samples of students from grades 4, 8, and 12. Suppose in 2001, the geography scores for male students had a mean of 264 with a standard deviation of 34. Assume that these scores are normally distributed with the given mean and standard deviation. The normal curve representing this distribution of the scores is symmetrical about a vertical line through The value 162 is under the curve between 162 and 366 is approximately below the mean, and the value 366 is Similarly, 95.44% of the male students scored between and above the mean. The area
The normal curve representing the distribution of the geography scores is symmetrical about the vertical line through the mean. Therefore, the vertical line through the mean score of 264 divides the curve into two symmetrical halves.
The value 162 is (264 - 102) below the mean, and the value 366 is (366 - 264) above the mean.
To find the area under the curve between 162 and 366, we need to standardize the values of 162 and 366 using the given mean and standard deviation. The standardized values are:
Z1 = (162 - 264) / 34 = -3
Z2 = (366 - 264) / 34 = 3
We can use a standard normal distribution table or a calculator to find the area under the curve between -3 and 3. The area under the curve between -3 and 0 is approximately 0.0013, and the area under the curve between 0 and 3 is also approximately 0.0013. Therefore, the total area under the curve between -3 and 3 is approximately:
0.0013 + 0.0013 = 0.0026
So the area under the curve between 162 and 366 is approximately 0.0026.
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