using the 98-95-99.7 rule, find the percentage of scores within 20 points of the mean of 50.
To find the percentage of scores within 20 points of the mean using the 98-95-99.7 rule, we need to calculate the z-scores first. The z-score measures the number of standard deviations a data point is from the mean.
1. Calculate the z-score for the upper bound (mean + 20):
z = (x - μ) / σ
z = (70 - 50) / σ
z = 20 / σ
2. Convert the z-score to a percentage using the normal distribution table or calculator.
According to the 98-95-99.7 rule, approximately 95% of the data falls within 2 standard deviations of the mean. This means that 2 standard deviations represent 95% of the data, and 1 standard deviation represents half of that (95% / 2 = 47.5%).
So, to find the percentage of scores within 20 points of the mean:
- Calculate the z-score for 20 points above the mean (upper bound).
- Find the corresponding percentage from the normal distribution table based on the z-score.
Keep in mind that the 98-95-99.7 rule is an estimation based on a normal distribution.