6. Area under the normal distribution Aa Aa The following figure shows the norma

Question

6. Area under the normal distribution Aa Aa The following figure shows the normal distribution with the proportion of the area under the normal curve contained within one, two, and three standard deviations of the mean. The last proportion on each side, 0.13%, depicts the remaining area under the curve. Specifically, 0.13% of the area under the standard normal distribution is located above z-score values greater than the mean (H) plus three standard deviations (+30). Also, because the normal distribution is symmetrical, 0.13% of the area under the standard normal distribution is located below z-score values less than the mean () minus three standard deviations (-30 34.13% 34.13% 13.59% 13.59% 2.15% 2.15% 0.13% 0.13% 30-20 Use the figure to help you answer the following questions 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 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 A score of 196 is below the mean, while a score of 332 is above the mean. This means that the percentage of male students with scores between 196 and 332 is A score of 366 is scores below 366 is above the mean. As a result, the percentage of male students with You can infer that 84.13% of the male students have scores above

Explanation / Answer

a)

(264 - 196)/34 = 2

hence 196 is 2 sd below the mean

(332-264)/34 = 68/34 = 2

hence 332 is 2 sd above the mean

% between 196 and 332 is 2 * 34.13 = 2(34.13 + 13.59) = 95.44 %

b)

(366 - 264)/34 = 102/34 = 3

hence 366 is 3 sd above the mean

hence % below 366 = 100 - 0.13 = 99.87

c)

84.13 % have scores have score which are 1 sd below mean

= 264 -34

= 230

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