Calcium levels in people are normally distributed with a mean of 9.7 mgdL and a standard deviation of 0.4 mgdL. Individuals with calcium levels in the bottom 15% of the population are considered to have low calcium levels. Find the calcium level that is the borderline between low calcium levels and those not considered low. Carry your intermediate computations to at least four decimal places. Round your answer to one decimal place.mgdL
Question
Calcium levels in people are normally distributed with a mean of 9.7 mgdL and a standard deviation of 0.4 mgdL. Individuals with calcium levels in the bottom 15% of the population are considered to have low calcium levels. Find the calcium level that is the borderline between low calcium levels and those not considered low. Carry your intermediate computations to at least four decimal places. Round your answer to one decimal place.mgdL
Solution
To solve this problem, we need to use the concept of z-scores in statistics. A z-score tells us how many standard deviations an element is from the mean.
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First, we need to find the z-score that corresponds to the bottom 15% of the population. We can use a z-table or a calculator with a normal distribution function for this. The z-score that corresponds to the bottom 15% of the population is approximately -1.0364.
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Next, we use the z-score formula to find the calcium level that corresponds to this z-score. The formula is:
Z = (X - μ) / σ
Where: Z is the z-score, X is the value we're looking for, μ is the mean, and σ is the standard deviation.
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We rearrange the formula to solve for X:
X = Zσ + μ
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Substituting the given values into the formula gives us:
X = -1.0364 * 0.4 + 9.7
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Calculating the above expression gives us X ≈ 9.2864 mg/dL.
So, the borderline between low calcium levels and those not considered low is approximately 9.3 mg/dL (rounded to one decimal place).
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