Answer:
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.
