The College Board originally scaled SAT scores so that the scores for each section were approximately normally distributed with a mean of 500 and a standard deviation of 100. Assuming scores follow a bell-shaped distribution, use the empirical rule to find the percentage of students who scored less than 400. a. 84% b. 16%

Question

Mathematics

Esmeralda Leon

26 December, 07:31

The College Board originally scaled SAT scores so that the scores for each section were approximately normally distributed with a mean of 500 and a standard deviation of 100. Assuming scores follow a bell-shaped distribution, use the empirical rule to find the percentage of students who scored less than 400. a. 84% b. 16%

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Marisa May · Answer 1

b. 16%

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 = 500

Standard deviation = 100

Percentage of students who scored less than 400:

400 = 500 - 1*100

So 400 is one standard deviation below the mean.

The normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are above.

Of those who are below, 68% are within 1 standard deviation of the mean, that is, between 400 and 500. So 100-68 = 32% are below 400.

0.5*0.32 = 0.16 = 16%

So the correct answer is:

b. 16%