Answer :
The relative frequency of rates less than 90 using the 68-95-99.7 rule is 84.13%.
The relative frequency of rates greater than 85 using the 68-95-99.7 rule is 31.73%.
The relative frequency of rates between 70 and 80 using the 68-95-99.7 rule is 47.72%.
Explanation: Given, Mean is 80, Standard deviation is 5.
Using 68-95-99.7 rule, we have 68% of the data will fall within one standard deviation of the mean. 95% of the data will fall within two standard deviations of the mean. 99.7% of the data will fall within three standard deviations of the mean. Using this, we need to calculate the relative frequency of rates.
A) To calculate the relative frequency of rates less than 90, we can calculate the Z-score as follows:
Z-score = (90 - 80) / 5
= 2
The Z-score is 2. This means that the heart rate of 90 is two standard deviations above the mean. Using the Z-table, we can find the proportion of scores below the Z-score of 2. The proportion is 0.9772. This means that the relative frequency of rates less than 90 using the 68-95-99.7 rule is 97.72%. The proportion of scores above 90 can be found by subtracting 0.9772 from 1. This proportion is 0.0228, or 2.28%.
B) To calculate the relative frequency of rates greater than 85, we can calculate the Z-score as follows:
Z-score = (85 - 80) / 5
= 1
The Z-score is 1. This means that the heart rate of 85 is one standard deviation above the mean. Using the Z-table, we can find the proportion of scores above the Z-score of 1. The proportion is 0.1587. This means that the relative frequency of rates greater than 85 using the 68-95-99.7 rule is 15.87%. The proportion of scores below 85 can be found by subtracting 0.1587 from 1. This proportion is 0.8413, or 84.13%.
C) To calculate the relative frequency of rates between 70 and 80, we can calculate the Z-scores for each value as follows:
Z-score1 = (70 - 80) / 5
= -2
Z-score2 = (80 - 80) / 5
= 0
The Z-score for 70 is -2, which means it is two standard deviations below the mean. The Z-score for 80 is 0, which means it is at the mean. Using the Z-table, we can find the proportion of scores between the Z-scores of -2 and 0. The proportion is 0.4772. This means that the relative frequency of rates between 70 and 80 using the 68-95-99.7 rule is 47.72%.
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