Answer :
The heartbeat values that separate the lower 27% from the rest are approximately 55.3 bpm, and the values that separate the upper 15% from the rest are approximately 66.6 bpm. These are calculated using the z-score formula with the given mean and standard deviation of the normal distribution of human heart rates.
The student is asking for the values that mark the threshold for the lower 27% and the upper 15% of human heart rates in a normal distribution with a mean of 59.1 bpm (beats per minute) and a standard deviation of 7.2 bpm. To find these values, we need to determine the z-scores that correspond to the 27th percentile and the 85th percentile (since the upper 15% would start at the 85th percentile) of a normal distribution. Then we can use the z-score formula to find the values for the heart rates.
For the lower 27%, assuming we look up the z-score in a standard normal distribution table, we would find a z-score about -0.61. Using the z-score formula:
X = \\mu + Z \\cdot \\sigma
where X is the value we want to find, \\mu is the mean, Z is the z-score, and \\sigma is the standard deviation. Plugging in the values, we get:
X = 59.1 + (-0.61) * 7.2 \\approx 55.3 bpm. Therefore the lower 27% of heart rates are below approximately 55.3 bpm.
For the upper 15%, the corresponding z-score would be close to 1.04. Again using the z-score formula, we get:
X = 59.1 + (1.04) * 7.2 \\approx 66.6 bpm. Therefore, the upper 15% of heart rates are above approximately 66.6 bpm.
