Question

A certain standardized survey has scores which range from 0 to 500, with decimal scores possible. Scores on the survey are normally distributed with a mean of 321 and a standard deviation of 42. What proportion of students taking the survey receive a score that is within 77 points of the mean? Round your answer to 4 decimal places.

Answer 1

To find the proportion of students receiving a score within 77 points of the mean, calculate the z-scores for the lower and upper bounds and find the area under the normal curve between these z-scores. The proportion is approximately 0.8892.

Step-by-step explanation:

To calculate the proportion of students receiving a score within 77 points of the mean, we need to find the z-scores for the lower and upper bounds and find the area under the normal curve between these z-scores.

The z-score for the lower bound is calculated as:

z = (score - mean) / standard deviation

z = (321 - 77 - 321) / 42

z = -1.83

The z-score for the upper bound is calculated as:

z = (score + mean) / standard deviation

z = (477 - 321) / 42

z = 3.71

Using a standard normal table or a calculator, we can find the area under the curve between -1.83 and 3.71, which represents the proportion of students receiving a score within 77 points of the mean. The calculated proportion is approximately 0.8892.