In a study of
500 men, diastolic blood pressure was found to be approximately normally distributed, with a mean of 82 millimeters (mm) of mercury and standard deviation of 10 millimeters.
Use the 68-95-99.7% rule to determine what percentage of the test group had a diastolic pressure between62 millimeters
and 102 millimeters of mercury.
95%
To determine the percentage of the test group that had a diastolic pressure between 62 and 102 millimeters of mercury, we need to use the 68-95-99.7% rule, also known as the empirical rule or the three sigma rule.
According to the empirical rule:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Given that the mean diastolic blood pressure is 82 millimeters of mercury and the standard deviation is 10 millimeters of mercury, we can calculate the range for one standard deviation, two standard deviations, and three standard deviations:
- One standard deviation: 82 ± (1 * 10) = 72 to 92
- Two standard deviations: 82 ± (2 * 10) = 62 to 102
- Three standard deviations: 82 ± (3 * 10) = 52 to 112
From the calculations, we can see that the diastolic pressure range of 62 to 102 millimeters of mercury falls within two standard deviations from the mean.
Using the empirical rule, we know that approximately 95% of the data falls within two standard deviations. Hence, approximately 95% of the test group had a diastolic pressure between 62 and 102 millimeters of mercury.
To use the 68-95-99.7% rule, we need to find the percentage of values within 1, 2, and 3 standard deviations from the mean.
Step 1: Calculate the z-scores for the lower and upper limits.
The z-score formula is:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
For the lower limit:
z_lower = (62 - 82) / 10
= -2
For the upper limit:
z_upper = (102 - 82) / 10
= 2
Step 2: Find the corresponding percentages for each z-score using the 68-95-99.7% rule.
For z = -2:
The percentage within 2 standard deviations on the left side of the mean is 0.15 (from 0.00 to 0.15).
For z = 2:
The percentage within 2 standard deviations on the right side of the mean is 0.15 (from 0.15 to 0.30).
Step 3: Calculate the percentage between the lower and upper limits.
The percentage between the lower and upper limits is the sum of the percentages within 2 standard deviations on each side of the mean:
Percentage = 0.15 + 0.15 = 0.30
Therefore, approximately 30% of the test group had a diastolic pressure between 62 and 102 millimeters of mercury.