Question

The Graduate Management Admission Test (GMAT) is a standardized exam used by many universities as part of the assessment for admission to graduate study in business. The average GMAT score is 547 (Magoosh website, January 5, 2015). Assume that GMAT scores are bell-shaped with a standard deviation of 100. Make sure to use the empirical rule to answer these questions. a. What percentage of GMAT scores are 647 or higher? % b. What percentage of GMAT scores are 747 or higher (to 1 decimal)? % c. What percentage of GMAT scores are between 447 and 547? % d. What percentage of GMAT scores are between 347 and 647 (to 1 decimal)? %

Answer 1

The empirical rule helps to determine the distribution of GMAT scores. Scores 647 or higher represent 16%, scores 747 or higher account for 2.5%, those between 447 and 547 represent 34%, and scores between 347 and 647 make up 81.5% of the distribution.

Step-by-step explanation:

The Graduate Management Admission Test (GMAT) scores are assumed to follow a bell-shaped distribution with a given mean and standard deviation. Using the empirical rule, we can answer the questions regarding the distribution of scores. The empirical rule states that:

• Approximately 68% of the data falls within one standard deviation of the mean.
• Approximately 95% of the data falls within two standard deviations of the mean.
• Approximately 99.7% of the data falls within three standard deviations of the mean.

For a mean GMAT score of 547 and a standard deviation of 100:

1. 647 or higher: This score is one standard deviation above the mean. Thus, approximately 16% (100% - 84%) of scores are 647 or higher since 84% of scores lie within one standard deviation above the mean.
2. 747 or higher: This score is two standard deviations above the mean. Therefore, about 2.5% (100% - 97.5%) of scores are 747 or higher since 97.5% of scores lie within two standard deviations above the mean.
3. Between 447 and 547: This range is within one standard deviation below the mean to the mean itself. Approximately 34% of scores are in this range.
4. Between 347 and 647: This range stretches from three standard deviations below the mean to one standard deviation above the mean. Around 81.5% (34% + 34% + 13.5%) of scores fall within this range.

Answer 2

a) 16% of GMAT scores are 647 or higher.

b) 2.5% of GMAT scores are 647 or higher.

c) 34% of GMAT scores are between 447 and 547.

d) 81.5% of GMAT scores are between 347 and 647.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 547

Standard deviation = 100

a. What percentage of GMAT scores are 647 or higher?

The Empirical rule states that 68% of the scores are within 1 standard deviation of the mean, that is, from 547 - 100 = 447 to 547 + 100 = 647. So 32% of the scores are outside the interval. Since the distribution is symmetric, 16% of them are lower than 447 and 16% of them are higher than 647.

So

16% of GMAT scores are 647 or higher.

b. What percentage of GMAT scores are 747 or higher (to 1 decimal)?

The Empirical rule states that 95% of the scores are within 2 standard deviations of the mean, that is, from 547 - 2*347 = 347 to 547 + 2*100 = 747. So 5% of the scores are outside the interval. Since the distribution is symmetric, 2.5% of them are lower than 347 and 2.5% of them are higher than 757

So

2.5% of GMAT scores are 647 or higher.

c. What percentage of GMAT scores are between 447 and 547?

447 is one standard deviation below the mean. The Empirical rule states that 68% of the scores are within 1 standard deviation of the mean, and since the distribution is symmetric, 34% are within one standard deviation below the mean and the mean, and 34% are within the mean and one standard deviation above the mean.

547 is the mean

447 is one standard deviation below the mean

So 34% of GMAT scores are between 447 and 547.

d. What percentage of GMAT scores are between 347 and 647 (to 1 decimal)?

The easist way is adding the percentage of scores from 347 to the mean(547) and the mean to 647.

Between 347 and 547

347 is two standard deviations below the mean. The Empirical rule states that 95% of the scores are within 2 standard deviations of the mean, and since the distribution is symmetric, 47.5% are within two standard deviation below the mean and the mean, and 47.5% are within the mean and two standard deviations above the mean.

So 47.5% of the scores are between 347 and 547

Between 547 and 647

447 is one standard deviation above the mean. The Empirical rule states that 68% of the scores are within 1 standard deviation of the mean, and since the distribution is symmetric, 34% are within one standard deviation below the mean and the mean, and 34% are within the mean and one standard deviation above the mean.

So 34% of the scores are between 547 and 647.

Between 347 and 647

47.5 + 34 = 81.5% of GMAT scores are between 347 and 647.