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 5.5 million cells per microliter and a standard deviation of 0.4 million cells per microliter.
(a) What is the minimum red blood cell count that can be in the top $30 \%$ of counts?
(b) What is the maximum red blood cell count that can be in the bottom $14 \%$ of counts?
(a) The minimum red blood cell count is million cells per microliter. (Round to two decimal places as needed.)
Answer
Final Answer: The minimum red blood cell count that can be in the top 30% of counts is approximately \(\boxed{5.71}\) million cells per microliter.
Step 1 ：We are given that the red blood cell counts for a population of adult males can be approximated by a normal distribution, with a mean of 5.5 million cells per microliter and a standard deviation of 0.4 million cells per microliter.
Step 2 ：We are asked to find the minimum red blood cell count that can be in the top 30% of counts. This is equivalent to finding the 70th percentile of the distribution.
Step 3 ：We use the properties of the normal distribution to find the z-score that corresponds to the 70th percentile. The z-score is a measure of how many standard deviations an element is from the mean.
Step 4 ：The z-score corresponding to the 70th percentile is approximately 0.524.
Step 5 ：We then convert this z-score to a red blood cell count using the given mean and standard deviation. The formula for this conversion is \(mean + z-score \times std\_dev\).
Step 6 ：Substituting the given values into the formula, we get \(5.5 + 0.524 \times 0.4 = 5.70976\).
Step 7 ：Rounding to two decimal places, we get 5.71.
Step 8 ：Final Answer: The minimum red blood cell count that can be in the top 30% of counts is approximately \(\boxed{5.71}\) million cells per microliter.
