It is also known that over a large number of 30

It is also known that over a large number of 30 - to 49 -year-old Caucasian women, their true mean SBP is normally distributed with mean $=120 \mathrm{mm}$ Hg and standard deviation $=14$ mm Hg. Also, over a large number of African American 30- to 49-year-old women, their true mean SBP is normal with mean $=130 \mathrm{mm} \mathrm{Hg}$ and standard deviation $=20 \mathrm{mm} \mathrm{Hg}$. Suppose we select a random $30-$ to 49 -year-old Caucasian woman and a random 30 - to 49 -year-old African American woman. What is the probability that the African American woman has a higher true SBP?

It is also known that over a large number of 30 - to 49 -year-old Caucasian women, their true mean SBP is normally distributed with mean $=120 \mathrm{mm}$ Hg and standard deviation $=14$ mm Hg. Also, over a large number of African American 30- to 49-year-old women, their true mean SBP is normal with mean $=130 \mathrm{mm} \mathrm{Hg}$ and standard deviation $=20 \mathrm{mm} \mathrm{Hg}$.
Suppose we select a random $30-$ to 49 -year-old Caucasian woman and a random 30 - to 49 -year-old African American woman. What is the probability that the African American woman has a higher true SBP?

Key Concepts

Standardization (Z-score)

Standardization involves converting a normal random variable into a standard normal variable with a mean of 0 and a standard deviation of 1 by subtracting the mean and dividing by the standard deviation. This process, resulting in a Z-score, simplifies the calculation of probabilities using standard normal distribution tables.

Independence of Variables

In probability theory, independence refers to situations where the outcome of one random variable does not affect the outcome of another. When two variables are independent, their joint behavior can be analyzed by considering the individual distributions separately.

Difference of Random Variables

When subtracting one independent normally distributed variable from another, the resulting random variable is also normally distributed. Its mean is the difference of the means, and its variance is the sum of the individual variances, which is key in comparing two independent normal variables.

Normal Distribution

A normal distribution is a common probability distribution that is symmetric and bell-shaped, characterized by its mean and standard deviation. It is used to model many natural phenomena where most outcomes cluster around the mean and the probabilities taper off symmetrically on both sides.

Mean and Standard Deviation

The mean represents the central value of the distribution, while the standard deviation measures the dispersion or variability around the mean. These parameters are essential in defining the shape and spread of the normal curve and are critical in determining probabilities.

Transcript

0:00 All right.

00:03 This problem is talking about heights of 20 to 29 year old females, and their mean height is 64 .1 inches, and the standard deviation is 2 .8 inches.

00:20 All right? so on all these problems, we are going to be using the normal distribution formula using z values.

00:31 That is the numbers we plug in is going to be a minus mu over the standard deviation.

00:41 So part a want to know about the percentile of a 60 inch tall girl.

00:49 All right.

00:51 So plug into the formula.

00:54 60 minus 64 .1 over 2 .8.

01:01 So that is negative 4 .1 over 2 .8, which gives you a z value of negative 1 .46.

01:14 All right.

01:15 Now, using a table of z values, i can look up negative 1 .46 and find that this is a percentage of 7 .22.

01:28 So a girl who is 60 inches tall is in the seventh percentile.

01:32 All right.

01:37 Letter b wants to know the same thing about a 70 inch tall girl.