The weights of crates of apples are normally distributed with a mean of 26.4 pounds and a standard deviation of 3.1 pounds. If a particular crate of apples weighs 31.6 pounds, we can find its percentile rank as follows:
First, we need to calculate the z-score of the crate's weight using the formula:
z = x − μ/ σ
where x is the weight of the crate, μ is the mean weight of all crates, and σ is the standard deviation of all crates.
Substituting the given values, we get:
z = 31.6 − 26.4/3.1
= 1.68
Next, we need to find the area under the standard normal distribution curve corresponding to the range of z-scores less than 1.68.
Using a z-table or statistical software, we find that this area is approximately 0.9535.
Finally, we convert this area to a percentile by multiplying by 100 and rounding to the nearest whole percent. Therefore, the percentile rank of the crate's weight is approximately 95%.