Use the empirical rule to solve the problem. The systolic blood pressure of 18 yr. old women is normally distributed with a mean of 120 mmHg (millimeters of Mercury) and a standard deviation of 14 mmHg. What is the percentage of 18 yr. old women that have a systolic blood pressure between 92 mmHg and 162 mmHg.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the two Z scores. Multiply by 100.
To use the empirical rule, which is also known as the 68-95-99.7 rule, we need three pieces of information: the mean, the standard deviation, and the assumption that the distribution is approximately normal.
Given:
Mean (μ) = 120 mmHg
Standard deviation (σ) = 14 mmHg
The empirical rule states that for a normal distribution:
- Approximately 68% of data falls within one standard deviation of the mean.
- Approximately 95% of data falls within two standard deviations of the mean.
- Approximately 99.7% of data falls within three standard deviations of the mean.
Now, to find the percentage of 18 yr. old women who have a systolic blood pressure between 92 mmHg and 162 mmHg, we need to calculate the z-scores for both values.
Z-score = (X - μ) / σ
For X = 92 mmHg:
Z1 = (92 - 120) / 14
For X = 162 mmHg:
Z2 = (162 - 120) / 14
Next, we can look up the corresponding areas under the standard normal distribution curve using the z-scores.
Using a z-table or a calculator, we can find the area (probability) for each z-score. The area between the two z-scores will give us the percentage of 18 yr. old women who have a systolic blood pressure between 92 mmHg and 162 mmHg.
Let's calculate the z-scores and find the area under the curve:
Z1 = (92 - 120) / 14
Z1 ≈ -2
Z2 = (162 - 120) / 14
Z2 ≈ 3
Using a standard normal distribution table or a calculator, we can find the area to the left of Z1, which is the area from negative infinity to Z1. Similarly, we can find the area to the left of Z2, which is the area from negative infinity to Z2.
Area (Z1) = 0.0228 (approximated from the Z-table)
Area (Z2) = 0.9987 (approximated from the Z-table)
To find the percentage of 18 yr. old women with systolic blood pressure between 92 mmHg and 162 mmHg, we need to subtract the area to the left of Z1 from the area to the left of Z2:
Percentage = Area (Z2) - Area (Z1)
Percentage = 0.9987 - 0.0228
Percentage ≈ 0.9759
Therefore, the percentage of 18 yr. old women that have a systolic blood pressure between 92 mmHg and 162 mmHg is approximately 97.59%.