Certainly! Let's solve the problem step-by-step.
### Given Information
- Average (mean) heart rate [tex]\( \mu \)[/tex] = 72 beats per minute
- Standard deviation [tex]\( \sigma \)[/tex] = 3 beats per minute
- We assume the heart rates are normally distributed.
### (a) Probability of having a heart rate less than 76 beats per minute
First, we need to calculate the z-score for 76 beats per minute.
Z-score Formula:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
where:
- [tex]\( X \)[/tex] is the value we are interested in (76 beats per minute)
- [tex]\( \mu \)[/tex] is the mean (72 beats per minute)
- [tex]\( \sigma \)[/tex] is the standard deviation (3 beats per minute)
Calculation:
[tex]\[ z = \frac{76 - 72}{3} = \frac{4}{3} = 1.3333 \][/tex]
Next, we use the z-score to find the corresponding probability from the standard normal distribution table.
The z-score of 1.3333 corresponds to a cumulative probability (area under the curve to the left of this z-score) of approximately 0.9088.
Probability:
[tex]\[ P(X < 76) = 0.9088 \][/tex]
This means there is a 90.88% chance that a randomly selected individual from this group will have a heart rate of less than 76 beats per minute.
### (b) Probability of having a heart rate higher than 70 beats per minute
Again, we first calculate the z-score for 70 beats per minute.
Calculation:
[tex]\[ z = \frac{70 - 72}{3} = \frac{-2}{3} = -0.6667 \][/tex]
Next, we use the z-score to find the corresponding cumulative probability from the standard normal distribution table.
The z-score of -0.6667 corresponds to a cumulative probability (area under the curve to the left of this z-score) of approximately 0.2525.
To find the probability of a heart rate higher than 70 beats per minute, we need the area to the right of this z-score:
Probability:
[tex]\[ P(X > 70) = 1 - P(X < 70) \][/tex]
[tex]\[ P(X > 70) = 1 - 0.2525 = 0.7475 \][/tex]
This means there is a 74.75% chance that a randomly selected individual from this group will have a heart rate of higher than 70 beats per minute.
### Summary
(a) The probability of a heart rate less than 76 beats per minute is approximately 90.88%.
(b) The probability of a heart rate higher than 70 beats per minute is approximately 74.75%.
