Questions: Calcium levels in people are normally distributed with a mean of 9.5 mg/dL and a standard deviation of 0.5 mg/dL. Individuals with calcium levels in the bottom 10% 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.

Calcium levels in people are normally distributed with a mean of 9.5 mg/dL and a standard deviation of 0.5 mg/dL. Individuals with calcium levels in the bottom 10% 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.

Transcript text: Calcium levels in people are normally distributed with a mean of $9.5 \frac{\mathrm{mg}}{\mathrm{dL}}$ and a standard deviation of $0.5 \frac{\mathrm{mg}}{\mathrm{dL}}$. Individuals with calcium levels in the bottom $10 \%$ 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.

Solution

Solution Steps

Step 1: Standardize the Problem

To find the calcium level corresponding to the 10% percentile, we first convert the percentile to a z-score. Using the standard normal distribution, the z-score corresponding to the 10% percentile is approximately -1.282.

Step 2: Convert Z-score to Raw Score

Using the formula $X = \mu + Z\sigma$, where:

$X$ is the calcium level we're solving for,
$\mu$ is the mean calcium level (9.5),
$Z$ is the z-score (-1.282), and
$\sigma$ is the standard deviation (0.5), we can find the specific calcium level. Substituting the values, we get $X = 9.5 + (-1.282)(0.5) = 8.9$.