**Week 5 Assignment Understanding the Empirical Rule**

**Question**

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.

**Question**

A random sample of small business stock prices has a sample mean of x¯=$54.82 and sample standard deviation of s=$8.95. Use the Empirical Rule to estimate the percentage of small business stock prices that are more than $81.67.

**Question**

A random sample of vehicle mileage expectancies has a sample mean of x¯=169,200 miles and sample standard deviation of s=19,400 miles. Use the Empirical Rule to estimate the percentage of vehicle mileage expectancies that are more than 188,600 miles.

**Question**

A random sample of hybrid vehicle fuel consumptions has a sample mean of x¯=53.2 mpg and sample standard deviation of s=4.8 mpg. Use the Empirical Rule to estimate the percentage of hybrid vehicle fuel consumptions that are less than 43.6 mpg.

**Solution:**

**Question**

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.

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