Questions: The red blood cell counts (in millions of cells per microliter) for a population of adult males can be approximated by a normal distribution, with a mean of 5.3 million cells per microliter and a standard deviation of 0.5 million cells per microliter. (a) What is the minimum red blood cell count that can be in the top 27% of counts? (b) What red blood cell counts would be considered unusual? (a) The minimum red blood cell count is million cells per microliter. (Round to two decimal places as needed.) (b) Red blood cell counts below million cells per microliter or above million cells per microliter would be considered unusual. (Round to two decimal places as needed.)

Solution Steps

To find the minimum red blood cell count that can be in the top \(27\%\) of counts, we need to determine the \(73^{rd}\) percentile of the normal distribution. Given the mean \(\mu = 5.3\) million cells per microliter and the standard deviation \(\sigma = 0.5\) million cells per microliter, we can use the z-score corresponding to the \(73^{rd}\) percentile.

The z-score for the \(73^{rd}\) percentile is approximately \(z_{0.73} \approx 0.610\). Thus, the minimum red blood cell count is calculated as follows:

\[ \text{Minimum RBC Count} = \mu + z_{0.73} \cdot \sigma = 5.3 + 0.610 \cdot 0.5 \approx 5.61 \text{ million cells per microliter} \]

Red blood cell counts are considered unusual if they fall more than \(2\) standard deviations from the mean. We calculate the lower and upper bounds for unusual counts as follows:

\[ \text{Lower Bound} = \mu - 2\sigma = 5.3 - 2 \cdot 0.5 = 4.30 \text{ million cells per microliter} \] \[ \text{Upper Bound} = \mu + 2\sigma = 5.3 + 2 \cdot 0.5 = 6.30 \text{ million cells per microliter} \]

Thus, red blood cell counts below \(4.30\) million cells per microliter or above \(6.30\) million cells per microliter would be considered unusual.

Final Answer