Answer :
Final answer:
This question is about the percentage of GMAT scores in specific ranges given mean and standard deviation. The answer involves calculating z-scores for given scores and using a Z-table to determine the percentage of scores within the ranges.
Explanation:
This question revolves around the concepts of mean, standard deviation, and z-score in the context of GMAT score distribution. (a) A GMAT score of 647 is 1 standard deviation above the mean as 647-547=100=1×standard deviation. To find the percentage of scores above 647, you need to look at a z-table. The value corresponding to z=1 is 0.8413, which means about 84.13% of scores fall below 647.
Hence, 100%-84.13% = 15.87% of GMAT scores are 647 or higher. Similarly, (b) A score of 747 is 2 standard deviations from the mean and according to the Z-table, 97.72% of scores fall below 747. So 100% - 97.72%=2.28% of GMAT scores are 747 or higher. (c) For scores between 347 and 547, we notice that these are -2 and 0 standard deviations from the mean, covering about 50% (from z=0) + 47.72% (from z=-2) = 97.72% of the distribution. (d) For scores between 447 and 747, these are -1 and 2 standard deviations from the mean. From a Z-table, these z-values correspond to 84.13% + (100%-97.72%) = 86.41% of the distribution.
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(a) Approximately 31.73% of GMAT scores are 647 or higher.
(b) Approximately 15.87% of GMAT scores are 747 or higher.
(c) Approximately 34.13% of GMAT scores are between 347 and 547.
(d) Approximately 68.26% of GMAT scores are between 447 and 747.
To solve these problems, we need to use the concept of z-scores, which measure the number of standard deviations a data point is from the mean. We can then use a standard normal distribution table or a calculator to find the corresponding percentages.
(a) To find the percentage of GMAT scores that are 647 or higher, we need to calculate the z-score for 647. The formula for the z-score is (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. In this case, x = 647, μ = 547, and σ = 100. Plugging these values into the formula, we get (647 - 547) / 100 = 1. Using a standard normal distribution table or calculator, we find that the area to the right of z = 1 is approximately 0.1587. To get the percentage, we multiply this by 100, resulting in approximately 15.87%. However, since we want the percentage for scores 647 or higher, we need to subtract this from 100, giving us approximately 100 - 15.87 = 84.13%.
(b) Similarly, to find the percentage of GMAT scores that are 747 or higher, we calculate the z-score for 747 using the same formula. With x = 747, μ = 547, and σ = 100, we have (747 - 547) / 100 = 2. Using the standard normal distribution table or calculator, we find that the area to the right of z = 2 is approximately 0.0228. Multiplying this by 100, we get approximately 2.28%. Subtracting this from 100, we find that approximately 100 - 2.28 = 97.72% of GMAT scores are below 747.
(c) To determine the percentage of GMAT scores between 347 and 547, we need to calculate the z-scores for both values. For 347, the z-score is (347 - 547) / 100 = -2, and for 547, the z-score is (547 - 547) / 100 = 0. Using the standard normal distribution table or calculator, we find the area to the right of z = -2 is approximately 0.9772 and the area to the right of z = 0 is 0.5. Subtracting these two values, we obtain approximately 0.9772 - 0.5 = 0.4772. Multiplying this by 100, we get approximately 47.72%.
(d) Finally, to find the percentage of GMAT scores between 447 and 747, we calculate the z-scores for both values. For 447, the z-score is (447 - 547) / 100 = -1, and for 747, the z-score is (747 - 547) / 100 = 2. Using the standard normal distribution table or calculator, we find the area to the right of z = -1 is approximately 0.8413, and the area to the right of z = 2 is approximately 0.0228. Subtracting these two values, we obtain approximately 0.8413 - 0.0228 = 0.8185. Multiplying this by 100, we get approximately 81.85%. However, since we want the percentage between the two scores, we subtract this from 100, resulting in approximately 100 - 81.85 = 18.15%.
In summary, approximately 31.73% of GMAT scores are 647 or higher, 15.87% are 747 or higher, 34.13% are between 347 and 547, and 68.26% are between 447 and 747.
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