Suppose that diastolic blood pressure readings of adult males have a bell-shaped distribution with a mean of 84
84
mmHg and a standard deviation of 9
9
mmHg. Using the empirical rule, what percentage of adult males have diastolic blood pressure readings that are greater than 102
102
mmHg? Please do not round your answer.
Suppose that diastolic blood pressure readings of adult males have a Bell-shaped distribution with a mean of 80 mmHg and a standard deviation of 10 mmHg. Using the empirical rule,what percentage of adult males have diastolic blood pressure readings that are less than 70 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 of the Z score. Multiply by 100 to get the percentage.
To find the percentage of adult males with diastolic blood pressure readings greater than 102 mmHg using the empirical rule, we need to calculate the z-score for 102 mmHg first.
The z-score formula is: z = (x - μ) / σ
Where:
x = observation (102 mmHg in this case)
μ = mean (84 mmHg)
σ = standard deviation (9 mmHg)
Now we can calculate the z-score:
z = (102 - 84) / 9
z = 18 / 9
z = 2
The empirical rule states that in a bell-shaped distribution:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
Since we have a z-score of 2, which is within 2 standard deviations of the mean, we can use the 95% approximation.
Therefore, the percentage of adult males with diastolic blood pressure readings greater than 102 mmHg is approximately 100% - 95% = 5%.
To find the percentage of adult males with diastolic blood pressure readings greater than 102 mmHg using the empirical rule, we need to determine how many standard deviations away from the mean 102 mmHg is.
The empirical rule states that for a bell-shaped distribution:
- Approximately 68% of the data falls within 1 standard deviation of the mean
- Approximately 95% of the data falls within 2 standard deviations of the mean
- Approximately 99.7% of the data falls within 3 standard deviations of the mean
First, we calculate the z-score, which measures how many standard deviations away from the mean a particular value is. The formula for the z-score is:
z = (X - μ) / σ
Where:
- X represents the value we want to find the position of (in this case, 102 mmHg)
- μ represents the mean of the distribution (84 mmHg)
- σ represents the standard deviation of the distribution (9 mmHg)
Let's calculate the z-score for 102 mmHg:
z = (102 - 84) / 9
z = 18 / 9
z = 2
The z-score tells us that 102 mmHg is 2 standard deviations away from the mean.
According to the empirical rule, approximately 95% of the data falls within 2 standard deviations of the mean. This means that approximately 2.5% of the data falls to the right side of the distribution beyond 2 standard deviations.
Therefore, the percentage of adult males with diastolic blood pressure readings greater than 102 mmHg is approximately 2.5%.
Note: This calculation assumes that the distribution follows a perfect bell-shaped curve and that the data is normally distributed. It's important to note that this is an estimation using the empirical rule, and the actual percentage may vary slightly depending on the specific distribution of the data.