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Answer:
Step-by-step explanation:
To determine the least amount of protein that places high school athletes in the middle 7.63% of the distribution, we need to find the protein intake values corresponding to the lower and upper bounds of the middle 7.63%. This can be done by determining the z-scores corresponding to the percentiles for the middle 7.63%.
Here are the steps:
Determine the z-scores for the middle 7.63% of a normal distribution:
The middle 7.63% implies that we are excluding 100% - 7.63% = 92.37% from the distribution.
This 92.37% is split equally on both sides of the middle. So, we have 46.185% in each tail.
Calculate the z-scores for these percentiles:
The lower percentile (left tail) is 46.185%.
The upper percentile (right tail) is 53.815% (since 50% + 3.815%).
To find these z-scores, we use the inverse of the cumulative distribution function (CDF) for a standard normal distribution.
Convert z-scores to protein intake values:
Use the z-score formula: x=μ+z⋅σx=μ+z⋅σ
Where μμ is the mean protein intake (116.94 grams).
σσ is the standard deviation (26.31 grams).
Let's perform these calculations:
Find the z-scores for the percentiles.
Convert these z-scores to actual protein intake values.
First, let's find the z-scores.
For 46.185%46.185%:
z1=norm.ppf(0.46185)z1=norm.ppf(0.46185)
For 53.815%53.815%:
z2=norm.ppf(0.53815)z2=norm.ppf(0.53815)
Now, convert these z-scores to protein intake values:
x1=μ+z1⋅σx1=μ+z1⋅σ
x2=μ+z2⋅σx2=μ+z2⋅σ
Since we need the least amount of protein, we focus on the lower bound:
x1=116.94+z1⋅26.31x1=116.94+z1⋅26.31
Let's calculate this.
The least amount of protein that places high school athletes in the middle 7.63% of the distribution is approximately 114.4202 grams. This is the minimum protein intake that ensures they are getting enough for their developing bodies.
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