Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 1 PROBABILITIES FOR CONTINUOUS RANDOM VARIABLES THE NORMAL DISTRIBUTION CHAPTER 8_B

Copyright © 2013, 2009, and 2007, Pearson Education, Inc.1

•PROBABILITIES FOR CONTINUOUS RANDOM

VARIABLES

•THE NORMAL DISTRIBUTION

CHAPTER 8_B


A continuous random variable has an infinite continuum of possible values in an interval.

Examples are: time, age, and size measures such as height and weight.

Continuous variables are usually measured in a discrete manner because of rounding.

REMINDER: Continuous Random Variable


A continuous random variable has possible values that form an interval. Its probability distribution is specified by a curve.

Each interval has probability between 0 and 1.

The interval containing all possible values has probability equal to 1.

Probability Distribution of a Continuous Random Variable


Figure 8.2 Probability Distribution of Commuting Time. The area under the curve for values higher than 45 is 0.15. Question: Identify the area under the curve represented by the probability that commuting time is less than 15 minutes, which equals 0.29.

Smooth curve approximation

Probability Distribution of a Continuous Random Variable


Useful Probability Relationships








Probabilities for Bell – Shaped Distributions or Continuous

Random Variables

Bell-Shaped Distributions


The normal distribution is symmetric, bell-shaped and characterized by its mean and standard deviation .

The normal distribution is the most important distribution in statistics.

Many distributions have an approximately normal distribution.

The normal distribution also can approximate many discrete distributions well when there are a large number of possible outcomes.

Many statistical methods use it even when the data are not bell shaped.

Normal Distribution


Normal distributions are Bell shaped Symmetric around the mean

The mean ( ) and the standard deviation ( ) completely describe the density curve.

Increasing/decreasing moves the curve along the horizontal axis.

Increasing/decreasing controls the spread of the curve.

Normal Distribution


Within what interval do almost all of the men’s heights fall? Women’s height?

Figure 8.4 Normal Distributions for Women’s Height and Men’s Height. For each different combination of and values, there is a normal distribution with mean and standard deviation . Question: Given that = 70 and = 4, within what interval do almost all of the men’s heights fall?

Normal Distribution


≈ 68% of the observations fall within one standard deviation of the mean.

≈ 95% of the observations fall within two standard deviations of the mean.

≈ 99.7% of the observations fall within three standard deviations of the mean.

Empirical Rule or 68-95-99.7 Rule for Any Normal Curve

Figure 8.5 The Normal Distribution. The probability equals approximately 0.68 within 1 standard deviation of the mean, approximately 0.95 within 2 standard deviations, and approximately 0.997 within 3 standard deviations. Question: How do these probabilities relate to the empirical rule?


Empirical Rule or 68 – 95 – 99.7% Rule


Heights of adult women can be approximated by a normal

distribution, inches; inches

68-95-99.7 Rule for women’s heights:

68% are between 61.5 and 68.5 inches

95% are between 58 and 72 inches

99.7% are between 54.5 and 75.5 inches

Example : 68-95-99.7% Rule

65 3.5

[ 65 3.5]

[ 2 65 2(3.5) 65 7]

[ 3 65 3(3.5) 65 10.5]


The z-score for a value x of a random variable is the number of standard deviations that x falls from the mean.

A negative (positive) z-score indicates that the value is below (above) the mean.

Z-scores can be used to calculate the probabilities of a normal random variable using the normal tables in the back of the book.

Z-Scores and the Standard Normal Distribution

zx


The formula for converting any value x to a z-score is

.

A z-score measures the number of standard deviations that a value falls from the mean

Standard Score (z – score)

x

zdeviation Standard

MeanValue


Shifting data: Adding (or subtracting) a constant to every data

value adds (or subtracts) the same constant to measures of position.

Adding (or subtracting) a constant to each value will increase (or decrease) measures of position: center, percentiles, max or min by the same constant.

Its shape and spread - range, IQR, standard deviation - remain unchanged.

Shifting and Rescaling Data


The following histograms show a shift from men’s actual weights to kilograms above recommended weight:



Rescaling data: When we multiply (or divide) all the data values

by any constant, all measures of position (such as the mean, median, and percentiles) and measures of spread (such as the range, the IQR, and the standard deviation) are multiplied (or divided) by that same constant.



The men’s weight data set measured weights in kilograms. If we want to think about these weights in pounds, we would rescale the data:



Examples: Midterm Exam II Practice Sheet


A standard normal distribution has mean and standard deviation .

When a random variable has a normal distribution and its values are converted to z-scores by subtracting the mean and dividing by the standard deviation, the z-scores follow the standard normal distribution.

Z-Scores and the Standard Normal Distribution

0 1


Table Z enables us to find normal probabilities. It tabulates the normal cumulative probabilities

falling below the point .

To use the table: Find the corresponding z-score. Look up the closest standardized score (z) in the

table. First column gives z to the first decimal place. First row gives the second decimal place of z.

The corresponding probability found in the body of the table gives the probability of falling below the z-score.

Table Z: Standard Normal Probabilities

z


Finding Probabilities Using The Standard Normal Table (Table Z)

The figure shows us how to find the area to the left when we have a z-score of 1.80:


Find the probability that a normal random variable takes a value less than 1.43 standard deviations above ;

Example: Using Table Z

( 1.43) 0.9236P z


Find the probability that a normal random variable assumes a value within 1.43 standard deviations of .

Probability below

Probability below


1.43 0.9236

1.43 0.0764

( 1.43 1.43) 0.9236 0.0764 0.8472P z


Figure 8.7 The Normal Cumulative Probability, Less than z Standard Deviations above the Mean. Table Z lists a cumulative probability of 0.9236 for , so 0.9236 is the probability less than 1.43 standard deviations above the mean of any normal distribution (that is, below ). The complement probability of 0.0764 is the probability above in the right tail.


1.43z

1.43 1.43


Sometimes we start with areas and need to find the corresponding z-score or even the original data value.

Example: What z-score represents the first quartile in a Normal model?

From Percentiles to z - Scores


Look in Table Z for an area of 0.2500.

The exact area is not there, but 0.2514 is pretty close.

This figure is associated with z = -0.67, so the first quartile is 0.67 standard deviations below the mean.

From Percentiles to z - Scores


To solve some of our problems, we will need to find the value of z that corresponds to a certain normal cumulative probability.

To do so, we use Table A in reverse.

Rather than finding z using the first column (value of z up to one decimal) and the first row (second decimal of z). Find the probability in the body of the table. The z-score is given by the corresponding

values in the first column and row.

How Can We Find the Value of z for a Certain Cumulative Probability?


Example: Find the value of z for a cumulative probability of 0.025.

Look up the cumulative probability of 0.025 in the body of Table A.

A cumulative probability of 0.025

corresponds to .

Thus, the probability that a normal

random variable falls at least

1.96 standard deviations

below the mean is 0.025.

How Can We Find the Value of z for a Certain Cumulative Probability?

1.96z


When you actually have your own data, you must check to see whether a Normal model is reasonable.Looking at a histogram of the data is a good way to check that the underlying distribution is roughly unimodal and symmetric. A more specialized graphical display that can help you decide whether a Normal model is appropriate is the Normal probability plot.If the distribution of the data is roughly Normal, the Normal probability plot approximates a diagonal straight line. Deviations from a straight line indicate that the distribution is not Normal.

Are You Normal? Normal Probability Plots


Nearly Normal data have a histogram and a Normal probability plot that look somewhat like this example:



A skewed distribution might have a histogram and Normal probability plot like below. In such cases it is unwise to use the Normal Model.



Z-scores can be used to compare observations from different normal distributions.

Picture the Scenario: There are two primary standardized tests used by

college admissions, the SAT and the ACT.

You score 650 on the SAT which has and and 30 on the ACT which has and .

How can we compare these scores to tell which score is relatively higher?

Example: Comparing Test Scores That Use Different Scales

500 100 21.0 4.7


Compare z-scores:

SAT:

ACT:

Since your z-score is greater for the ACT, you performed relatively better on this exam.

Using Z-scores to Compare Distributions

z650 500

1001.5

z30 21

4.71.91


If we’re given a value x and need to find a probability, convert x to a z-score using , use a table of normal probabilities (or software, or a calculator) to get a cumulative probability and then convert it to the probability of interest

If we’re given a probability and need to find the value of x , convert the probability to the related cumulative probability, find the z-score using a normal table (or software, or a calculator), and then evaluate .

SUMMARY: Using Z-Scores to Find Normal Probabilities or Random Variable x Values

( ) /z x

x z

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Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 1 PROBABILITIES FOR CONTINUOUS RANDOM VARIABLES THE NORMAL DISTRIBUTION CHAPTER 8_B