Math 225N Week 5 Assignment: Understanding
the Empirical Rule
1. A random sample of CO2 levels in a school has a sample mean of x¯=598.4 ppm and
sample standard deviation of s=86.7 ppm. Use the Empirical Rule to determine the
approximate percentage of CO2 levels that lie between 338.3 and 858.5 ppm.
To use the Empirical Rule, we need to know how many standard deviations
from the mean are the given values 338.3 and 858.5.
Since the mean is 598.4, we see that the value 338.3 is
598.4−338.3=260.1 ppm below the mean. This is 3 standard
deviations, since 260.1=3×86.7, so 338.3 is 3 standard deviations less
than the mean.
Similarly, the value 858.5 is 858.5−598.4=260.1 ppm above the
mean. Again, this is 3 standard deviations, since 260.1=3×86.7, so
858.5 is 3 standard deviations greater than the mean.
The Empirical Rule states that approximately 99.7% of the data is within 3
standard deviations of the mean. So by the Empirical Rule, we can say that
approximately 99.7% of CO2 levels in the school are between 338.3 and
858.5 ppm.
2. Suppose that a random sample of redwood trees has a sample mean
diameter of x¯=24.1 feet, with a sample standard deviation of s=3.7 feet.
Since the diameters of redwood trees are generally symmetric and bell-
shaped, we can apply the Empirical Rule.
Between what two diameters are approximately 68% of the data?
he value 1 standard deviation below the mean is
x¯−s=24.1−3.7=20.4.
, The value 1 standard deviation above the mean is
x¯+s=24.1+3.7=27.8.
So by the Empirical Rule, we can say that approximately 68% of the
redwood diameters are between 20.4 and 27.8 feet.
3. Suppose a random sample of monthly rainfalls in a given area has a
sample mean of x¯=22.2 inches, with a sample standard deviation of
s=3.5 inches. Since rainfall amounts in this area are generally symmetric
and bell-shaped, we can apply the Empirical Rule.
Between what two amounts are approximately 99.7% of the data?
The value 3 standard deviations below the mean is
x¯−3s=22.2−3(3.5)=11.7.
The value 3 standard deviations above the mean is
x¯+3s=22.2+3(3.5)=32.7.
So by the Empirical Rule, we can say that approximately 99.7% of the
rainfall amounts are between 11.7 and 32.7 inches.
4. Suppose a random sample of adult women has a sample mean height of
x¯=64.3 inches, with a sample standard deviation of s=2.4 inches. Since
height distribution are generally symmetric and bell-shaped, we can apply
the Empirical Rule.
Between what two heights are approximately 99.7% of the data?
The Empirical Rule states that approximately 99.7% of the data is within
three standard deviations of the mean. In terms of the sample mean x¯ and
sample standard deviation s, that is between x¯−3s and x¯+3s.
The value three standard deviations below the mean is
the Empirical Rule
1. A random sample of CO2 levels in a school has a sample mean of x¯=598.4 ppm and
sample standard deviation of s=86.7 ppm. Use the Empirical Rule to determine the
approximate percentage of CO2 levels that lie between 338.3 and 858.5 ppm.
To use the Empirical Rule, we need to know how many standard deviations
from the mean are the given values 338.3 and 858.5.
Since the mean is 598.4, we see that the value 338.3 is
598.4−338.3=260.1 ppm below the mean. This is 3 standard
deviations, since 260.1=3×86.7, so 338.3 is 3 standard deviations less
than the mean.
Similarly, the value 858.5 is 858.5−598.4=260.1 ppm above the
mean. Again, this is 3 standard deviations, since 260.1=3×86.7, so
858.5 is 3 standard deviations greater than the mean.
The Empirical Rule states that approximately 99.7% of the data is within 3
standard deviations of the mean. So by the Empirical Rule, we can say that
approximately 99.7% of CO2 levels in the school are between 338.3 and
858.5 ppm.
2. Suppose that a random sample of redwood trees has a sample mean
diameter of x¯=24.1 feet, with a sample standard deviation of s=3.7 feet.
Since the diameters of redwood trees are generally symmetric and bell-
shaped, we can apply the Empirical Rule.
Between what two diameters are approximately 68% of the data?
he value 1 standard deviation below the mean is
x¯−s=24.1−3.7=20.4.
, The value 1 standard deviation above the mean is
x¯+s=24.1+3.7=27.8.
So by the Empirical Rule, we can say that approximately 68% of the
redwood diameters are between 20.4 and 27.8 feet.
3. Suppose a random sample of monthly rainfalls in a given area has a
sample mean of x¯=22.2 inches, with a sample standard deviation of
s=3.5 inches. Since rainfall amounts in this area are generally symmetric
and bell-shaped, we can apply the Empirical Rule.
Between what two amounts are approximately 99.7% of the data?
The value 3 standard deviations below the mean is
x¯−3s=22.2−3(3.5)=11.7.
The value 3 standard deviations above the mean is
x¯+3s=22.2+3(3.5)=32.7.
So by the Empirical Rule, we can say that approximately 99.7% of the
rainfall amounts are between 11.7 and 32.7 inches.
4. Suppose a random sample of adult women has a sample mean height of
x¯=64.3 inches, with a sample standard deviation of s=2.4 inches. Since
height distribution are generally symmetric and bell-shaped, we can apply
the Empirical Rule.
Between what two heights are approximately 99.7% of the data?
The Empirical Rule states that approximately 99.7% of the data is within
three standard deviations of the mean. In terms of the sample mean x¯ and
sample standard deviation s, that is between x¯−3s and x¯+3s.
The value three standard deviations below the mean is
