The per capita consumption of coffee by people in the United States in a recent year was normally distributed, with a mean of 24.2 gallons and a standard deviation of 8.1 gallons.
What proportion of people in the U.S. drink more than 20 gallons of coffee a year?
Use z-scores.
Formula:
z = (x - mean)/sd
x = 20
mean = 24.2
sd = 8.1
Plug the values into the formula and calculate z. Then check a z-table using the z-score for the proportion of people who drink more than 20 gallons.
I hope this will help get you started.
To find the proportion of people in the U.S. who drink more than 20 gallons of coffee a year, we can use the cumulative distribution function (CDF) of the normal distribution.
1. Standardize the value of 20 gallons using the formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
Plugging in the values:
z = (20 - 24.2) / 8.1
= -0.5185
2. Look up the standardized value (-0.5185) in a standard normal distribution table or use a statistical calculator to find the cumulative probability associated with that value.
The CDF representing the proportion of people who drink more than 20 gallons of coffee is equal to 1 minus the cumulative probability of the standardized value.
3. Subtract the cumulative probability from 1 to find the proportion of people who drink more than 20 gallons of coffee.
P(X > 20) = 1 - P(Z < -0.5185)
Using a standard normal distribution table or a statistical calculator, we find that the cumulative probability associated with -0.5185 is 0.2975.
Therefore, the proportion of people in the U.S. who drink more than 20 gallons of coffee a year is 1 - 0.2975 = 0.7025, or 70.25%.