If systolic blood pressure in a population has a mean of 120 mmHg and a standard deviation of 10 mmHg, what percentage of people have systolic blood pressure between 110 mmHg and 1... If systolic blood pressure in a population has a mean of 120 mmHg and a standard deviation of 10 mmHg, what percentage of people have systolic blood pressure between 110 mmHg and 130 mmHg?
Understand the Problem
The question is asking for the percentage of people within a given range of systolic blood pressure using the normal distribution. The mean is 120 mmHg with a standard deviation of 10 mmHg. We need to find the cumulative percentage of the population that has blood pressure between 110 mmHg and 130 mmHg.
Answer
Approximately 68.26% of people have systolic blood pressure between 110 mmHg and 130 mmHg.
Answer for screen readers
Approximately 68.26% of people have systolic blood pressure between 110 mmHg and 130 mmHg.
Steps to Solve
- Standardize the values
To find the percentage of people with blood pressure between 110 mmHg and 130 mmHg, first convert these values into z-scores using the formula: $$ z = \frac{x - \mu}{\sigma} $$ where ( \mu = 120 ) mmHg (mean) and ( \sigma = 10 ) mmHg (standard deviation).
For 110 mmHg: $$ z_1 = \frac{110 - 120}{10} = -1 $$
For 130 mmHg: $$ z_2 = \frac{130 - 120}{10} = 1 $$
- Look up the z-scores
Now, find the cumulative probabilities for the z-scores from the standard normal distribution table or using a calculator.
For ( z_1 = -1 ): The cumulative probability ( P(Z < -1) \approx 0.1587 ).
For ( z_2 = 1 ): The cumulative probability ( P(Z < 1) \approx 0.8413 ).
- Calculate the percentage between the two values
To find the percentage of the population with blood pressure between 110 mmHg and 130 mmHg, subtract the lower cumulative probability from the upper cumulative probability: $$ P(110 < X < 130) = P(Z < 1) - P(Z < -1) $$
Calculating the difference: $$ P(110 < X < 130) = 0.8413 - 0.1587 = 0.6826 $$
- Convert to percentage
To express this as a percentage: $$ 0.6826 \times 100 \approx 68.26% $$
Approximately 68.26% of people have systolic blood pressure between 110 mmHg and 130 mmHg.
More Information
The result falls within the empirical rule, which states that about 68% of values lie within one standard deviation of the mean in a normal distribution.
Tips
- Forgetting to standardize: Always ensure to convert raw values to z-scores before using normal distribution tables.
- Misreading the table: Double-check cumulative probabilities for z-scores, as negative values indicate areas to the left of the mean.
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