Suppose the weekly use of gasoline for motor travel by adults in North America is approximately normally distributed, with a mean of 14 gallons and a standard deviation of 6 gallons. What proportion of adults use more than 18 gallons per week?
To find the proportion of adults who use more than 18 gallons of gasoline per week, we need to calculate the area under the normal curve to the right of 18 gallons. Here's how you can do it step by step:
Step 1: Standardize the value of 18 gallons.
To standardize a value, subtract the mean and divide by the standard deviation. In this case, let's call the standardized value "z".
z = (x - mean) / standard deviation
where x is the value you want to standardize, mean is the mean of the distribution, and standard deviation is the standard deviation of the distribution.
In our case, the value of interest is 18 gallons, the mean is 14 gallons, and the standard deviation is 6 gallons.
z = (18 - 14) / 6 = 4 / 6 = 0.67
Step 2: Find the proportion to the right of 18 gallons.
We need to find the area under the normal curve to the right of the standardized value of 0.67. This represents the proportion of adults who use more than 18 gallons per week.
To find this proportion, we can use a standard normal distribution table or a calculator that provides the cumulative distribution function (CDF) for the standard normal distribution.
Using either method, we find the proportion to the right of 0.67 is approximately 0.2525.
Therefore, approximately 25.25% of adults use more than 18 gallons of gasoline per week.