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The amount of protein consumed by high school athletes is normally distributed with a mean of 116.94 grams and a standard deviation of 26.31 grams. If the amount of protein consumed by high school athletes is in the middle 7.63%, they are getting enough for their developing bodies. What is the least amount of protein and still have enough? Round all answers to 4 decimal places where possible.

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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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