First we need to understand the range we are talking about here (4.2 to 5.4) in terms of standardized scores. Then we can use what we know about the normal distribution to estimate the percentage of women who fall in that range.
One way to think about this problem is to try to understand how many standard deviation units separate 4.2 and 5.4 from the mean of the sample. The first calculations here will be pretty simple.
(lower bound - mean)/standard deviation and (upper bound - mean)/standard deviation
So we get (4.2 - 4.577)/.382 and (5.4 - 4.577)/.382
Complete the computations and you get:
-0.987 and 2.154. There are a number of tables out that provide the kind of percentages you would like to know to complete the next step. They are all based on the normal distribution and require that you know the z score (i.e., standardized score) of interest. That is essentially what I just calculated in the previous step (the deviation from the mean divided by the sample's standard deviation). Many introductory statistics texts have these sorts of tables in the appendices. You can also find them online.
Using such a table we find that a standardized score of -0.987 is at the 16.2 percentile and a standardized score 2.15 is at the 98.4 percentile.
I wish I could draw out the normal curve at this stage because it would help you understand the next step. So what you really want to know is about the percentage of women between these numbers. One way to approach answering that question is to calculate the percentage of women between the mean and each of the boundaries and add those percentages together to get the total number of women who fall within that range.
We know that one of the properties of the normal distribution is that 50 percent of cases fall above the mean and 50 percent fall below the mean. So a value that is at the mean represents the 50th percentile exactly.
Now because the lower boundary (and its percentile) are below the mean, to find the percent of women with scores between 4.2 and 4.577 (i.e., scores between the lower boundary and the mean) we subtract 16.2% from 50% and get 33.8%.
Because the upper boundary is above the mean the process is a little different in that we subtract 50% from 98.4% and get 48.4%.
As a last step add these two percentages together and you will get 33.8% + 48.4% = 82.2%, which is the percent of women whose blood cell counts fall between 4.2 and 5.4.
I hope that helps.