Questions: The graph illustrates the distribution of test scores taken by College Algebra students. The maximum possible score on the test was 130, while the mean score was 79 and the standard deviation was 12. What is the approximate percentage of students who scored between 79 and 115 on the test? % What is the approximate percentage of students who scored between 79 and 91 on the test? % What is the approximate percentage of students who scored higher than 103 on the test? What is the approximate percentage of students who scored lower than 43 on the test?

Solution Steps

The z-score is calculated as (x - mean) / standard deviation. The mean is 79 and the standard deviation is 12.

• For x = 115: z = (115 - 79) / 12 = 3
• For x = 91: z = (91 - 79) / 12 = 1
• For x = 103: z = (103 - 79) / 12 = 2
• For x = 43: z = (43 - 79) / 12 = -3

The empirical rule, or 68-95-99.7 rule, states that approximately 68% of the values fall within 1 standard deviation of the mean, approximately 95% of the values fall within 2 standard deviations of the mean and approximately 99.7% of the values fall within 3 standard deviations of the mean for normal distribution.

• Between 79 and 115: This range represents 3 standard deviations above the mean (z=3). Half of 99.7% is 49.85%.
• Between 79 and 91: This range represents 1 standard deviation above the mean (z=1). Half of 68% is 34%.
• Higher than 103: This is 2 standard deviations above the mean. The area above is given by (100 - 95)/2 which is 2.5%

Final Answer: