Question

Cholesterol levels for women aged 20 to 34 follow an approximately normal distribution with mean 185 milligrams per deciliter (mg/dl). Women with cholesterol levels above 220 mg/dl are considered to have high cholesterol and about 18.5% of women fall into this category. What is the standard deviation of the distribution of cholesterol levels for women aged 20 to 34?

Answer 1

Answer:

The standard deviation for women aged 20 to 34 is approximately 39mg/dl.

Step-by-step explanation:

The key to solve this question is using the formula for z-scores, that is, the transformed values for any normally distributed data in order to consult the probabilities related with these values using the standard normal distribution table. We can obtain a z-score using the formula:

$\\ z = \frac{x - \mu}{\sigma}$

Where

x is a normally distributed value to transform in a z-score.

$\\ \mu\;is\;the\;population\;mean$.

$\\ \sigma\;is\;the\;population\;standard\;deviation$.

We have from the question that:

The population mean is:

$\\ \mu = 185\frac{mg}{dl}$

The probability for those women with cholesterol levels above 220mg/dl is 18.5%. Mathematically, P(x>220mg/dl) = 0.185.

For using the cumulative standard normal distribution table, we need to determine the probability for those values below 220mg/dl, that is, P(x<220mg/dl), which is:

$\\ P(x<220\frac{mg}{dl}) = 1 - P(x>220\frac{mg}{dl}) = 1 - 0.185 = 0.815$.

This means that for P(x<220mg/dl) = 0.815, that is, approximately, 81.5% of the cases are below this value. The question that arises here is to determine the z-score that corresponds to this probability.

Consulting the cumulative standard normal table, the value of z for P(z<0.815) is, approximately between z = 0.89 (0.8133) and z =0.90 (0.8159).

Considering the value of z = 0.90, we can determine an approximate value for the standard deviation from the formula for z-scores as follows (without using units mg/dl):

$\\ z = \frac{x - \mu}{\sigma}$

$\\ 0.90 = \frac{220 - 185}{\sigma}$

$\\ \sigma = \frac{220 - 185}{0.90}$

$\\ \sigma = 38.88888\dots\approx 38.89$

Following the same steps, for z = 0.89, the value for the population standard deviation is 39.33. So, having only approximation from the standard normal table, we can say that the standard deviation is between 38.89 and 39.33.

As a result, we can say that an approximate value for the population standard deviation for the cholesterol levels for women aged 20 to 34 is about 39mg/dl.

The graph below shows a normal distribution with mean = 185mg/dl and a standard deviation = 39mg/dl, whose shaded area is for values P(x<220mg/dl) = 0.8152, or representing 81.52% of the cases or a P(x>220) = 1 - 0.8152 = 0.1848 (very near to the probability given in the question P(x>220) = 0.185).