Response:
a) 16% of GMAT scores are at least 647 or more.
b) 2.5% of GMAT scores are at least 647 or more.
c) 34% of GMAT scores fall between 447 and 547.
d) 81.5% of GMAT scores are between 347 and 647.
Detailed explanation:
The Empirical Rule indicates that for a normally distributed variable:
68% of values lie within 1 standard deviation of the mean.
95% of values lie within 2 standard deviations of the mean.
99.7% of values lie within 3 standard deviations of the mean.
In this scenario, we have:
Mean = 547
Standard deviation = 100
a. What percentage of GMAT scores are at least 647 or more?
The Empirical rule suggests that 68% of scores fall within 1 standard deviation from the mean, specifically from 547 - 100 = 447 to 547 + 100 = 647. Consequently, 32% lie outside this range. Given symmetry in the distribution, 16% are below 447, and 16% exceed 647.
This means
16% of GMAT scores are 647 or greater.
b. What percentage of GMAT scores are 747 or higher (to one decimal place)?
The Empirical rule states that 95% of scores fall within 2 standard deviations of the mean (from 547 - 2*100 = 347 to 547 + 2*100 = 747), implying that 5% of scores lie beyond this range. Due to the distribution's symmetry, 2.5% are below 347, and 2.5% exceed 747.
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2.5% of GMAT scores are 647 or greater.
c. What percentage of GMAT scores are between 447 and 547?
Since 447 is within one standard deviation beneath the average, the Empirical rule indicates that 68% of scores are within one standard deviation of the mean. As a result, 34% fall between one standard deviation below the mean and the mean, and another 34% are between the mean and one standard deviation above it.
Hence, 34% of GMAT scores are between 447 and 547.
d. What percentage of GMAT scores are between 347 and 647 (to one decimal)?
The simplest approach is to aggregate percentages of scores from 347 to 547 and from 547 to 647.
Between 347 and 547:
347 is two standard deviations beneath the mean. The Empirical rule states 95% of scores fall within two standard deviations of the mean. Because of symmetry, 47.5% are within two standard deviations below the mean, and 47.5% are above the mean.
Thus, 47.5% are between 347 and 547.
Between 547 and 647:
Since 447 is one standard deviation above the mean, and based on the Empirical rule, an additional 34% are between the mean and one standard deviation above the mean.
Consequently, the sum of scores between 347 and 647 totals 81.5%.
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