The annual per capita consumption of fresh apples (in pounds) in a nearby state can be approximated by a normal distribution, with a mean of 15.4 pounds and a standard deviation of 3.8 pounds. (a) What is the smallest annual per capita consumption of apples that can be in the top 25% of consumption
To find the smallest annual per capita consumption of apples that can be in the top 25% of consumption, we need to find the value that corresponds to the 75th percentile of the normal distribution.
Here's how you can calculate it:
1. Determine the z-score corresponding to the desired percentile. The z-score represents the number of standard deviations away from the mean a particular value is. Since we are interested in the top 25% (which corresponds to the upper tail of the distribution), we need to find the z-score for the 75th percentile. You can use a standard normal distribution table or a calculator with a normal distribution function to find this value. In this case, the z-score for the 75th percentile is approximately 0.674.
2. Use the z-score formula to solve for the value corresponding to the desired percentile. The z-score formula is given by:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the value we are looking for
- μ is the mean
- σ is the standard deviation
Rearranging the formula, we get:
x = μ + z * σ
Plugging in the given values:
x = 15.4 + 0.674 * 3.8
3. Calculate the value for x:
x ≈ 15.4 + 2.562
x ≈ 17.962
So, the smallest annual per capita consumption of apples that can be in the top 25% of consumption is approximately 17.962 pounds.