Answer:
1) Percentages of scores less than 100: 50%
2) Percentage of scores less than 140: 97.5%
3) Percentage of scores less than 80: 16%
4) Percentage of scores between 80 and 120: 68%
5) Percentage of scores between 80 and 140: 81.5%
1) Percentage of rates less than 70 : 50%
2) Percentage of rates less than 55 : 16%
3) Percentage of rates less than 85 : 97.5%
4) Percentage of rates greater than 85 : 2.5%
5) Percentage of rates greater than 55 : 84%
6) Percentage of rates between 55 and 100: 81.5%
7) Percentage of rates between 70 and 100: 47.5%
Step-by-step explanation:
We have a random variable normally distributed with a mean of 100 and a standard deviation of 20.
1) Percentages of scores less than 100: 50%
As the mean is 100, 50% of the data lies below 100.
2) Percentage of scores less than 140: 97.5%
The data is what lies below (mean +2 sd). In this case, applies the 95% rule for the higher scores (above 100), which means we have 95/2=47.5 of the data between 100 and 140.
The data below 100 represents 50%.
So the scores under 140 are 50+47.5=97.5%.
3) Percentage of scores less than 80: 16%
The scores under 80 are a (mean-1 sd).
This means that is half of the data, less 68/2=34 (the area that is under the first standard deviation of the mean and the mean).
Then, the scores under 80 are 50-34=16%.
4) Percentage of scores between 80 and 120: 68%
The scores are under one deviation of the mean (to both sides). The 68% rule applies.
5) Percentage of scores between 80 and 140: 81.5%
The lower scores, between 80 and 100 are in the area between one deviation and the mean, so it has a percentage of 68/2=34%.
The higher scores are 2 deviations frome the mean, so they have 95/2=47.5% of the scores.
Between 80 and 140 are 34+47.5=81.5% of the scores.
