Question

A certain standardized test's math scores have a bell-shaped distribution with a mean of 520 and a standard deviation of 105. Complete parts (a) through (c).

(a) What percentage of standardized test scores is between 205 and 835? % (Round to one decimal place as needed.)
(b) What percentage of standardized test scores is less than 205 or greater than 835? % (Round to one decimal place as needed.)
(c) What percentage of standardized test scores is greater than 730? 1% (Round to one decimal place as needed.) Enter your answer in each of the answer boxes. 12,005 MAR 8

Answer 1

To find the percentage of standardized test scores between 205 and 835, calculate the z-scores for each value and use the standard normal distribution table. Subtract the area to the left of -3.24 from the area to the left of 3.00. To find the percentage less than 205 or greater than 835, subtract the percentage between 205 and 835 from 100%. To find the percentage greater than 730, calculate the z-score for 730 and subtract the area to the left of 2.00 from 100%.

Step-by-step explanation:

To find the percentage of standardized test scores between 205 and 835, we need to calculate the z-scores for each of these values and then use the standard normal distribution table. The z-score for 205 is calculated as (205 - 520) / 105 ≈ -3.24, and the z-score for 835 is calculated as (835 - 520) / 105 ≈ 3.00. Looking up these z-scores in the standard normal distribution table, we find that the area to the left of -3.24 is approximately 0.0006, and the area to the left of 3.00 is approximately 0.9987. To find the percentage between these two scores, we subtract the area to the left of -3.24 from the area to the left of 3.00: 0.9987 - 0.0006 ≈ 0.9981, which is approximately 99.8%. Therefore, approximately 99.8% of standardized test scores are between 205 and 835.

To find the percentage of standardized test scores less than 205 or greater than 835, we can subtract the percentage between 205 and 835 from 100%: 100% - 99.8% ≈ 0.2%. Therefore, approximately 0.2% of standardized test scores are less than 205 or greater than 835.

To find the percentage of standardized test scores greater than 730, we need to calculate the z-score for 730 and use the standard normal distribution table. The z-score for 730 is calculated as (730 - 520) / 105 ≈ 2.00. Looking up this z-score in the standard normal distribution table, we find that the area to the left of 2.00 is approximately 0.9772. To find the percentage greater than 730, we subtract the area to the left of 2.00 from 100%: 100% - 0.9772 ≈ 99.0%. Therefore, approximately 99.0% of standardized test scores are greater than 730.