Heights of men on a baseball team have a bell-shaped distribution with a mean of 169 cm169 cm and a standard deviation of 9 cm9 cm. Using the empirical rule, what is the approximate percentage of the men between the following values?
a. % of the men are between 142 cm and 196 cm.
(Round to one decimal place as needed.)
b. % of the men are between 151 cm and 187 cm.
(Round to one decimal place as needed.)
My answer
A) 99%
B)68%
To use the empirical rule, you need to understand the concept of standard deviations. According to the empirical rule, 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.
a. To find the percentage of men between 142 cm and 196 cm, you need to calculate how many standard deviations each value is away from the mean:
Step 1: Calculate the number of standard deviations for 142 cm:
(142 cm - 169 cm) / 9 cm = -3
Step 2: Calculate the number of standard deviations for 196 cm:
(196 cm - 169 cm) / 9 cm = 3
Since the range from -3 to 3 covers approximately 99.7% of the data, we can estimate that the percentage of men between 142 cm and 196 cm is also approximately 99.7%.
b. Similarly, to find the percentage of men between 151 cm and 187 cm, apply the same steps:
Step 1: Calculate the number of standard deviations for 151 cm:
(151 cm - 169 cm) / 9 cm = -2
Step 2: Calculate the number of standard deviations for 187 cm:
(187 cm - 169 cm) / 9 cm = 2
Since the range from -2 to 2 covers approximately 95% of the data, we can estimate that the percentage of men between 151 cm and 187 cm is approximately 95%.
Therefore, the approximate percentages of men between the given values are:
a) Approximately 99.7%
b) Approximately 95%