Heights of men on a baseball team have a bell-shaped distribution with a mean of 169 cm and a standard deviation of 9 cm. Using the empirical rule, what is the approximate percentage of the men between the following values? a. 151 cm and 187 cm. b. 160 cm and 178 cm.
Please help I do not understand how to compute this! Thank you for your assistance.
Z = (score-mean)/SD
Do you know the 68-95-99.7 rule? Approximately 68% of scores in normal distribution are within one standard deviation (Z)(34% on each side of the mean), 95% within 2 SD, and 99.7% within 3 SD.
To compute the approximate percentage of men between two values using the empirical rule, also known as the 68-95-99.7 rule, you need to understand the concept of standard deviations.
The empirical rule states that for a bell-shaped distribution, approximately:
- 68% of the data falls within one standard deviation of the mean,
- 95% of the data falls within two standard deviations of the mean,
- 99.7% of the data falls within three standard deviations of the mean.
Using this information, let's calculate the approximate percentage of men between the given values:
a. To find the percentage of men between 151 cm and 187 cm, we need to determine how many standard deviations each value is from the mean.
First, subtract the mean from each value:
151 cm - 169 cm = -18 cm
187 cm - 169 cm = 18 cm
Next, divide the difference by the standard deviation:
-18 cm / 9 cm = -2
18 cm / 9 cm = 2
Based on the empirical rule, we know that approximately 95% of the data falls within two standard deviations of the mean. Since the values -2 and 2 fall within this range, the approximate percentage of men between 151 cm and 187 cm is also 95%.
b. Following the same steps, let's calculate the approximate percentage of men between 160 cm and 178 cm:
160 cm - 169 cm = -9 cm
178 cm - 169 cm = 9 cm
-9 cm / 9 cm = -1
9 cm / 9 cm = 1
Again, based on the empirical rule, we know that approximately 68% of the data falls within one standard deviation of the mean. Since the values -1 and 1 fall within this range, the approximate percentage of men between 160 cm and 178 cm is 68%.
So, the approximate percentage for a. is 95% and for b. is 68%.