Question

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 0.4 million cells per microliter.
(a) what is the minimum red blood cell count that can be in the top 26% 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.)

Explanation:

Step1: Find the z - score for the top 26%

The top 26% means an area of 0.74 to the left of the value we want. Looking up 0.74 in the standard - normal distribution table, the z - score $z\approx0.64$.

Step2: Use the z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 5.3$ (mean), $\sigma=0.4$ (standard deviation), and we want to find $x$. Rearranging the formula for $x$ gives $x=\mu + z\sigma$.
Substitute $\mu = 5.3$, $z = 0.64$, and $\sigma = 0.4$ into the formula: $x=5.3+0.64\times0.4$.

Step3: Calculate the value of $x$

$x=5.3 + 0.256=5.56$

For part (b), in a normal distribution, unusual values are outside of the range $\mu\pm2\sigma$.
Lower limit: $x_1=\mu - 2\sigma=5.3-2\times0.4=5.3 - 0.8 = 4.5$.
Upper limit: $x_2=\mu + 2\sigma=5.3+2\times0.4=5.3 + 0.8 = 6.1$.
Unusual red - blood cell counts are less than 4.5 million cells per microliter or greater than 6.1 million cells per microliter.

Answer:

(a) 5.56
(b) Less than 4.5 million cells per microliter or greater than 6.1 million cells per microliter.