A pharmaceutical study measures the resting heart rate of 200 participants and finds that the data is bell-shaped with a mean of 80 beats per minute and a standard deviation of 5 b... A pharmaceutical study measures the resting heart rate of 200 participants and finds that the data is bell-shaped with a mean of 80 beats per minute and a standard deviation of 5 beats per minute. According to the Empirical Rule, what range will include approximately 95% of the participants' resting heart rates?
Understand the Problem
The question is asking to apply the Empirical Rule (68-95-99.7 rule) to find the range that includes approximately 95% of the participants' resting heart rates, given the mean and standard deviation.
Answer
The range is $70$ to $90$ beats per minute.
Answer for screen readers
The range that includes approximately 95% of the participants' resting heart rates is from 70 to 90 beats per minute.
Steps to Solve
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Identify the Mean and Standard Deviation The mean resting heart rate is given as ( \mu = 80 ) beats per minute, and the standard deviation is ( \sigma = 5 ) beats per minute.
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Apply the Empirical Rule According to the Empirical Rule, approximately 95% of the data falls within 2 standard deviations from the mean.
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Calculate the Range for 95% To find the range that includes 95% of the participants’ heart rates:
- Calculate the lower limit: $$ \text{Lower Limit} = \mu - 2\sigma = 80 - 2(5) = 80 - 10 = 70 $$
- Calculate the upper limit: $$ \text{Upper Limit} = \mu + 2\sigma = 80 + 2(5) = 80 + 10 = 90 $$
- Determine the Final Range The final range that includes approximately 95% of the participants' resting heart rates is from 70 to 90 beats per minute.
The range that includes approximately 95% of the participants' resting heart rates is from 70 to 90 beats per minute.
More Information
This result is based on the Empirical Rule, which applies to normally distributed data. The rule states that about 68% of values lie within one standard deviation, about 95% lie within two standard deviations, and about 99.7% lie within three standard deviations from the mean.
Tips
- Miscalculating the range by not multiplying the standard deviation by 2.
- Confusing the standard deviation with the mean.
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