Because serum cholesterol is related to age and sex, some...

Because serum cholesterol is related to age and sex, some investigators prefer to express it in terms of $z$ -scores. If $X=$ raw serum cholesterol, then $$Z=\frac{X-\mu}{\sigma}$$, where $\mu$ is the mean and $\sigma$ is the standard deviation of serum cholesterol for a given age-gender group. Suppose $Z$ is regarded as a standard normal random variable. What is $\operatorname{Pr}(-1.0<Z<1.5) ?$

Key Concepts

Standard Normal Distribution

A standard normal distribution is a special case of the normal distribution that has a mean of zero and a standard deviation of one. It is widely used in statistics because many variables can be normalized to this form, and it has well-tabulated properties and probabilities.

Z-Score Standardization

This is a method used to convert a raw score into a standard score by subtracting the mean and then dividing by the standard deviation. The result represents the number of standard deviations a value is from the mean, which allows for comparison across different datasets or groups.

Cumulative Distribution Function (CDF)

The CDF for a random variable gives the probability that the variable will take a value less than or equal to a given number. For a standard normal distribution, the CDF is used to determine the probability that the variable falls within a specific range by calculating the difference between the CDF values at the endpoints of the interval.

Transcript

00:01 So we're looking at a standardized score, which is, again, a number minus the mean divided by a standard deviation.

00:06 It's kind of the essence of statistics.

00:08 And they talked about the cholesterol being given in that form for the serum.

00:13 And we want to look at how do we find this probability.

00:17 Well, i'm going to draw you a diagram, first of all.

00:20 Here is the standard normal distribution, that good old bell -shaped curve.

00:24 It's supposed to be symmetrical, centered about a mean of zero.

00:28 And with the standard deviation of 1, so this would be a z score of 1, z score of 2, z score of 3, and usually we don't draw it out any further than negative 3 to 3, because this range includes 99 .7 % of the values.

00:45 And so we have your situation, what's the likelihood of having the probability be between negative 1 and negative 1 .5? so we want basically that area, and the area underneath the standard, curve is one.

01:01 So we want to find that area.

01:02 You know, and you can eyeball it and guesstimate, but that's not what we would do here.

01:06 So the first thing you do is if you're using a table in the back of a book or in your book, you're going to find the area below 1 .5.

01:14 And so you'll go to your table in your book and you'll look up, get my pencil back on the screen.

01:24 You're going to look up 1 .5 and then up here is 0 .0.

01:28 And you'll look up.

01:28 And you you're going to find this number.

01:30 And that number when i look at my book is 0 .9332...