Answer :
We can conclude that approximately 2.5% of students are likely to get a score below 275 on the GRE.
According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean, 95% of the data falls within two standard deviations of the mean, and 99.7% of the data falls within three standard deviations of the mean.
Given that the mean GRE score is 500 and the standard deviation is 75, we can calculate that a score of 275 is 2.33 standard deviations below the mean
z-score = (275-500) ÷75 = -2.33.
Using the empirical rule, we can determine that approximately 2.5% of students are expected to score lower than 275 on the GRE. This is because the area under the normal distribution curve beyond 2.33 standard deviations below the mean (i.e., to the left of z-score = -2.33) is about 0.01, and this area is multiplied by 2 to account for both tails of the distribution, giving a total probability of 0.025 or 2.5%.
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The correct answer is that approximately 25% of students are likely to get a score below 275.
To solve this problem, we can apply the 68–90 5–90 9.7 empirical rule, which states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within tree standard deviations.
Given that the mean GRE score is 500 and the standard deviation is 75, we can calculate the following:
- One standard deviation below the mean: (500 - 75 = 425)
- Two standard deviations below the mean: [tex]\(500 - 2 \times 75 = 350\)[/tex]
- Three standard deviations below the mean: [tex]\(500 - 3 \times 75 = 275\)[/tex]
According to the empirical rule, about 99.7% of the scores will be above three standard deviations below the mean. Since we are looking for the percentage of students likely to get a score below 275, we are interested in the remaining percentage that is not within the three standard deviations range.
The percentage of scores that are within three standard deviations of the mean is 99.7%. Therefore, the percentage of scores that are below three standard deviations below the mean (275) is:
[tex]\(100\% - 99.7\% = 0.3\%\)[/tex]
However, this 0.3% represents the percentage of scores below the mean minus three standard deviations. Since the GRE score distribution is symmetric, the same percentage (0.15%) will be above the mean plus three standard deviations. Therefore, to find the percentage below 275, we need to consider only half of the 0.3%:
[tex]\(0.3\% / 2 = 0.15\%\)[/tex]
To convert this percentage to a more common format, we can express it as a fraction of the total percentage:
[tex]\(0.15\% = \frac{0.15}{100} = \frac{15}{10000} = \frac{3}{2000}\)[/tex]
To find the percentage of students likely to get a score below 275, we subtract this fraction from 50% (since 50% of students will score below the mean, and we want to find those scoring below 275, which is three standard deviations below the mean):
[tex]\(50\% - \frac{3}{2000} \approx 50\% - 0.00015 \approx 49.9985\%\)[/tex]
Converting this to a percentage gives us approximately 49.9985%, which we can round to 50% for practical purposes. Since we are considering only the scores below the mean, we take half of this percentage to get the percentage of students scoring below 275:
[tex]\(50\% / 2 = 25\%\)[/tex]
