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About what percent of the area under the curve of the standard normal distribution is between z = -2.576 and z = 2.576 (or within 2.576 standard deviations of the mean)?

A) 99.7%
B) 95%
C) 90%
D) 80%

User Licysca
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1 Answer

5 votes

Final answer:

Option A). The percentage of the area under the standard normal distribution curve between z-scores of -2.576 and 2.576 is more than 95% but less than 99.7%. The closest option provided is 99.7%, although the actual value is closer to 99%.

Step-by-step explanation:

The percentage of the standard normal distribution's area under the curve that is between z-scores -2.576 and 2.576 is the subject of the question. We know that around 68% of the values reside within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean, based on the empirical rule, also known as the 68-95-99.7 rule.

It is evident that the percentage of the area under the curve we are looking for is greater than 95% but less than 99.7% because the z-scores in question, -2.576 and 2.576, are outside the z-scores of -2 and 2, which correspond to 95% of the area under the curve, but inside the z-scores of -3 and 3, which correspond to 99.7% of the area. Because the percentage that falls between z-scores of -2.576 and 2.576 must be slightly greater than the 95% that falls between z-scores of -2 and 2, being closer to 99% based on traditional z-tables, the correct answer is not available in the options provided. However, the closest choice given is Option A) 99.7%.

User Mahmoud Moawad
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