heights of men on a baseball team have a bell shaped distribution with a mean of 172 cm and a distribution of 9cm. Using the empirical rule, what is the approximate percentage of the men between the following values?
a. ___% of the men are between 145cm and 199cm
b.___% of the men are between 154cm and 190cm
I think you mean standard deviation of 9
http://davidmlane.com/hyperstat/z_table.html
To find the approximate percentage of men between the given values using the empirical rule, we need to calculate the z-scores for the given values and then use the z-table to find the corresponding percentages.
The empirical rule states that for a bell-shaped distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% falls within two standard deviations of the mean.
- Approximately 99.7% falls within three standard deviations of the mean.
Given: Mean (μ) = 172 cm and Standard Deviation (σ) = 9 cm
a. To find the percentage of men between 145 cm and 199 cm:
First, we need to calculate the z-scores for these values using the formula:
z = (x - μ) / σ
For 145 cm:
z1 = (145 - 172) / 9 = -3
For 199 cm:
z2 = (199 - 172) / 9 = 3
Now, we can use the z-table (Standard Normal Distribution table) to find the percentage of data between these two z-scores.
From the z-table, we find that the area under the curve between -3 and 3 is approximately 0.9973.
To convert this to a percentage, we multiply by 100:
Percentage between 145 cm and 199 cm ≈ 0.9973 x 100 ≈ 99.73%
Therefore, approximately 99.73% of the men are between 145 cm and 199 cm.
b. To find the percentage of men between 154 cm and 190 cm:
Again, we need to calculate the z-scores for these values:
For 154 cm:
z1 = (154 - 172) / 9 ≈ -2
For 190 cm:
z2 = (190 - 172) / 9 ≈ 2
Using the z-table, we find that the area under the curve between -2 and 2 is approximately 0.9545.
Converting this to a percentage:
Percentage between 154 cm and 190 cm ≈ 0.9545 x 100 ≈ 95.45%
Therefore, approximately 95.45% of the men are between 154 cm and 190 cm.