Final answer:
The Empirical Rule states that approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. Using this rule, we can determine the range of study hours for a given percentage of students and find the percentage of students below a certain value.
Step-by-step explanation:
The Empirical Rule states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
a) Since we know that 68% of the data falls within one standard deviation, we can conclude that 68% of the students have study hours that are between one standard deviation below the mean and one standard deviation above the mean.
b) To find the percentage of students with study hours below 23, we need to find the z-score for 23 using the formula z = (X - mean) / standard deviation. Then, we can use a standard normal distribution table or calculator to find the percentage of students below that z-score. The final answer is approximately 84%.