A particular fruit's weights are normally distributed, with a mean of 764 grams and a standard deviation of 33 grams.
The heaviest 15% of fruits weigh more than how many grams?
Give your answer to the nearest gram.
To determine the weight in grams that represents the heaviest 15% of fruits, we need to find the z-score associated with this percentile and then convert it back to the original weight using the mean and standard deviation.
Step 1: Find the z-score associated with the percentile of 15%.
Since the weights are normally distributed, we can use the z-table or a calculator to find the z-score corresponding to the 15th percentile. The z-score for a 15% right-tail probability is approximately -1.04.
Step 2: Convert the z-score back to the original weight.
We can use the formula z = (x - μ) / σ, where z is the z-score, x is the original weight, μ is the mean, and σ is the standard deviation.
Rearranging the formula to solve for x, we have x = z * σ + μ.
Substituting the values, we have:
x = -1.04 * 33 + 764 = -34.32 + 764 ≈ 729.68 grams.
Rounding to the nearest gram, the heaviest 15% of fruits weigh more than approximately 730 grams.
To find the weight in grams that corresponds to the heaviest 15% of fruits, we need to use the cumulative distribution function (CDF) of a normal distribution.
1. First, we need to determine the z-score corresponding to the desired cumulative probability (15%). The z-score represents the number of standard deviations a value is away from the mean in a normal distribution.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
x = the value
μ = the mean
σ = the standard deviation
For the heaviest 15% of fruits, we want to find the z-score that corresponds to a cumulative probability of 0.85 (100% - 15% = 85%).
2. Next, we need to use a standard normal distribution table or a calculator to find the z-score that corresponds to a cumulative probability of 0.85.
Looking up this value in a standard normal distribution table or using a z-score calculator, we find that the z-score is approximately 1.036.
3. Now that we have the z-score, we can calculate the weight in grams.
Using the formula:
x = μ + (z * σ)
Substituting the given values, we have:
x = 764 + (1.036 * 33)
x = 764 + 34.188
x ≈ 798.188
Rounding to the nearest gram, the heaviest 15% of fruits weigh more than approximately 798 grams.
the upper 15% of weights are a little over one S.D. above the mean
so, more than ... 764 + 33
using a z-score table ... 1.036 S.D
so, more than ... 764 + (33 * 1.036)