a. According to the empirical rule,
for a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean,
approximately 95% falls within two standard deviations of the mean,
and approximately 99.7% falls within three standard deviations of the mean.
In this case, the mean is 258.1 and the standard deviation is 68.1. So, three standard deviations from the mean would be:
258.1 - 3(68.1) = 53.8
and
258.1 + 3(68.1) = 462.4
Therefore, approximately 99.7% of women have platelet counts between 53.8 and 462.4.
b. To find the percentage of women with platelet counts between 190.0 and 326.2, we need to standardize these values using the formula:
z = (x - μ) / σ
where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.
For the lower bound of 190.0:
z = (190.0 - 258.1) / 68.1 = -1.001
For the upper bound of 326.2:
z = (326.2 - 258.1) / 68.1 = 1.001
Using a standard normal distribution table or calculator, we can find the area under the curve between -1.001 and 1.001, which represents the percentage of women with platelet counts between 190.0 and 326.2. This area is approximately 68.2%.
Therefore, approximately 68.2% of women have platelet counts between 190.0 and 326.2.