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Answer :
To solve this problem, we use the properties of the normal distribution since GMAT scores are said to be bell-shaped.
The average GMAT score (mean, [tex]\mu[/tex]) is 550, and the standard deviation ([tex]\sigma[/tex]) is 100.
To find the percentage of scores for various ranges, we can use the concept of the standard normal distribution (Z-scores). The Z-score tells us how many standard deviations away a particular score is from the mean.
(a) What percentage of GMAT scores are 650 or higher?
First, we calculate the Z-score for 650.
[tex]Z = \frac{650 - \mu}{\sigma} = \frac{650 - 550}{100} = 1[/tex]
From standard normal distribution tables, a Z-score of 1 corresponds to the area of 0.8413 to the left. Therefore, the area to the right (which represents the scores 650 or higher) is:
[tex]1 - 0.8413 = 0.1587[/tex]
So, approximately 15.87% of GMAT scores are 650 or higher.
(b) What percentage of GMAT scores are 750 or higher?
Next, we calculate the Z-score for 750.
[tex]Z = \frac{750 - \mu}{\sigma} = \frac{750 - 550}{100} = 2[/tex]
For a Z-score of 2, the area to the left is 0.9772. Therefore, the area to the right is:
[tex]1 - 0.9772 = 0.0228[/tex]
So, approximately 2.28% of GMAT scores are 750 or higher.
(c) What percentage of GMAT scores are between 450 and 550?
Calculate the Z-scores for 450 and 550.
For 450:
[tex]Z = \frac{450 - \mu}{\sigma} = \frac{450 - 550}{100} = -1[/tex]
For 550 (which is the mean, so Z = 0):
[tex]Z = 0[/tex]
The area between Z = -1 and Z = 0 is 0.3413 (from the symmetry property of the normal distribution).
So, approximately 34.13% of GMAT scores are between 450 and 550.
(d) What percentage of GMAT scores are between 350 and 650?
Calculate the Z-scores for 350 and 650.
For 350:
[tex]Z = \frac{350 - \mu}{\sigma} = \frac{350 - 550}{100} = -2[/tex]
For 650:
[tex]Z = 1[/tex]
The area between Z = -2 and Z = 1 is the area to the left of 1 minus the area to the left of -2.
From standard normal distribution tables:
- Area to the left of Z = 1 is 0.8413
- Area to the left of Z = -2 is 0.0228
So, the area between them is:
[tex]0.8413 - 0.0228 = 0.8185[/tex]
Thus, approximately 81.85% of GMAT scores are between 350 and 650.
In summary, using the properties of the normal distribution and Z-scores allows us to find the percentage of scores in certain ranges by calculating and using the area under the standard normal curve.
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