Mathematics, 07.04.2020 21:10 CodyJ4503

The centers for disease control and prevention reported that diastolic blood pressures of adult women in the us are approximately normally distributed with a mean 80.5 and standard deviation 9.9 what proportion of women have blood pressures lower than 70

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The proportion of women that have blood pressures lower than 70 is 0.14457 or $\\ P(z$ . That is, about 14.457% of women have blood pressures lower than 70.

Step-by-step explanation:

We have normally distributed data here. The normal distribution is defined by its parameters, namely, the mean (80.5, in this case) and the standard deviation (9.9, in this case).

To find the probabilities associated with the normal distribution, we can use the standard normal distribution (with mean = 0, and standard deviation = 1), and more precisely, the cumulative standard normal distribution. All values for this distribution are z-scores, which are defined by the formula:

$\\ z = \frac{x - \mu}{\sigma}$ [1]

Where

$\\ z$ is the z-score.

$\\ x$ is the "raw" value for which we want to find the corresponding z-score.

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

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

Roughly speaking, a z-score represents the distance from the mean in standard deviations units. Negative values are scores below the mean, whereas positive values, scores above the mean.

Well, we can use the formula [1], and then, with this z-score, consult any standard normal table (available on the Internet or in every Statistics books) to find the probability related to it.

We already know that the family of the normal distributions are symmetrical, which is important when finding negative z-scores since most tables contain positive values for z-scores.

The proportion of women that have blood pressures lower than 70

First step: Find the z-score for the raw value x = 70.

Using the formula [1], we have:

$\\ z = \frac{70 - 80.5}{9.9}$

$\\ z = \frac{-10.5}{9.9}$

$\\ z = -1.0606...\approx -1.06$

This value of z tells us that the raw value of x = 70 is represented by a standardized value of z = -1.06, which says that this value is -1.06 standard deviations units from the mean (that is, below the mean). This value is negative. As a result, we can consult the cumulative standard normal table for z = 1.06. After that, and because normal distributions are symmetrical around the mean (as we explain it before), we have:

$\\ P(z$ [2]

Second step: Find the corresponding probability of P(z<1.06) using the cumulative standard normal table, and then, use formula [2].

$\\ P(z$

Then

$\\ P(z1.06)$

The value for $\\ P(z$ is the probability we are finding for. Thus, the proportion of women that have blood pressures lower than 70 is 0.14457 or $\\ P(z$ .

Thus, we can say that, approximately, 14.457% of women have blood pressures lower than 70.

The graph below represents this proportion of women that have blood pressures lower than 70 [P(x<70)], for a normal distribution with $\\ \mu = 80.5$ and $\\ \sigma = 9.9$ .