Chapter 2 Section 2
ALL normal distributions are the same if we measure in units of size σ
about the mean µ as center
Because of this…
Standardizing, changing to units
If x is an observation from a distribution that has mean µ and
Chapter 2 Section 2 Continued…
What Does A Z-Score Tell Us…
How many standard deviations the original observation falls away
from the mean, and in which direction
Standard Normal Distribution, is a normal distribution of N(0,1) N(0,1) ~ mean µ of 0 and standard deviation σ
Note: If a variable x has any normal distribution N(µ , σ) then
What is the z-score of the
value indicated on the curve?
What is the z-score of the
value indicated on the curve?
How do you use this?
The mean score on the SAT is 1500, with
a standard deviation of 240. The ACT, a
different college entrance examination,
has a mean score of 21 with a standard
deviation of 6.
If Bobby scored 1740 on the SAT and
Bobby
Kathy
z
=
1
Kathy scored higher.
Kathy’s z-score shows that she scored
1.5 standard deviations above the
mean.
Bobby only scored 1 standard
deviation above the mean.
Chapter 2 Section 2 Continued…
Normal Distribution Calculations
An area under a density curve is a proportion of the
observations in a distribution
ALL normal distributions are the SAME when we standardize Area of any normal curve can be found from the Two (really is one)
z-tables, both Positive and Negative Z-scores
For each Z-VALUE there is an assigned area under the curve to
Chapter 2 Section 2 Continued…
Positive Z-scores
Chapter 2 Section 2 Continued…
Finding Proportion GIVEN A Value For “Normal Curve”
Step 1~ State the problem in terms of the observed variable x. Draw a PICTURE of the distribution and SHADE the area of interest under the curve
Step 2 ~ Standardize x to restate the problem in terms of a standard normal variable z. Make sure to state the the Z-Score on the figure
Step 3 ~ Find the required area under the standard normal curve, Using the Z-table (Table A) for both positive and negative. Know that the total area under the curve is 1 or 100%
Step 4 ~ Write your conclusion in the context of the problem
Chapter 2 Section 2 Continued…
Normal distribution is an approximation, NOT a description for
every detail in the actual data
Note: the proportion of as well as . There
is no area under and exactly over as well as over and exactly over
Goal, is to sketch the area you want, match the area with that
of the one the table gives you!
What if…we made a z-value that falls outside the range
covered by table A??
There is a very little area under the standard normal curve outside
the range covered by Table A. Therefore, this area is known as an area of ZERO
?
Chapter 2 Section 2 Continued…
Chapter 2 Section 2 Continued…
Finding A Value GIVEN a Proportion For “Normal Curve”
We may want to find an observed value with a given
proportion of the observations above or below
To do this we would read the Table-A Or (Z-Table) backward
1st Find the given proportion on the table, construct the
Chapter 2 Section 2 Continued…
Example: Finding Proportion GIVEN A Value For “Normal Curve”
&
Chapter 2 Section 2 Continued…
Not By Hand…Through The Calculator
2nd Vars Distr
normalcdf (lower bound, upper bound, µ, σ) 1E 99 …used for upper bound in special cases
-1E 99 … used for lower bound in special cases
What is the deal with normalpdf (x, µ, σ) X is a single observation
Helps you find inflection points and draw normal curve USE normalcdf Always!!
invnorm (percent of area to left, , )
Chapter 2 Section 2 Continued…
Accessing Normality
Method 1~ Constructing a Frequency Histogram Or A Stemplot Look to see the graph is approximately symmetric and bell-shaped
about the mean
Method 2 ~ Construct a Normal Probability Plot The plotted points should lie close
Use 2nd Stat Plot…example
ANY normal distribution produces a straight line on the plot
Homework Question, 1.20 (c)
Typically for a histogram one would construct the y-axis in
terms of individual frequencies of our individual intervals. How ever, cumulative frequency takes the frequency of the first
then adds the it to the second then divides the totally.
Deeper Look into Standard Deviation
Definition of deviation: the amount by which a single
measurement differs from a fixed value
In our case the fixed value is the population mean
Describes the distribution in terms of the mean
Deeper definition ~ Provides an indication of how far the
individual responses to a question very or “deviate” from the population mean..
It tells the researcher how spread out the responses are, example,
are the responses concentrated around the mean, or scattered far and wide Example 2: Did all of your respondents rate your
“product” in the middle of the scale, or did other feel different, how different
