Question

For all U.S. students who take the ACT, ACT scores are normally distributed with an average

ACT composite score of 20.75 and a standard deviation of 5.88. Using the Standard Deviation Rule
and the given information, 16% of ACT scores for all U.S. students are less than what value of an
ACT score?

Answer 1

Approximately 16% of U.S. students taking the ACT have a composite score of less than 14.87, which is calculated by subtracting one standard deviation (5.88) from the mean score (20.75).

Step-by-step explanation:

According to the Standard Deviation Rule, or the empirical rule, for a normal distribution, about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations. Knowing this, we can infer that approximately 16% of the data is either above the mean plus one standard deviation or below the mean minus one standard deviation. In this case, because we are interested in ACT scores that are less than a certain value, we would look at the lower end of the distribution.

Given that the mean ACT composite score is 20.75 and the standard deviation is 5.88, 16% of scores lie below the mean minus one standard deviation. To find the score below which 16% of test scores lie, we calculate:

Mean - Standard Deviation = 20.75 - 5.88 = 14.87

Therefore, approximately 16% of U.S. students have an ACT composite score of less than 14.87.

Answer 2

To find the ACT score that corresponds to the 16th percentile, you can use the Standard Deviation Rule. The ACT score that is less than 16% of all U.S. students is 15.55.

Step-by-step explanation:

To find the ACT score that corresponds to the 16th percentile, we can use the Standard Deviation Rule. The first step is to calculate the z-score using the formula: z = (x - mean) / standard deviation. In this case, the mean is 20.75 and the standard deviation is 5.88. We need to find the z-score for the 16th percentile, which is -0.94. Next, we can use the z-score formula to solve for x: x = (z * standard deviation) + mean. Plugging in the values, we get x = (-0.94 * 5.88) + 20.75 = 15.55.