## (Solved) The Normal Distribution and Probability The Normal Distribution and Probability Program Transcript JENNIFER ANN MORROW: Welcome to the Normal...

I unlocked this question and saw you already answered it but it wasn'tÂ properly answered. It was not detailed. You repeated the question you were asked multiple times without answering the question. That is why i am asking the question again. Pls BEWARE OF PLAGIARISM. View attachment for transcript of video

For this Assignment, review the interactive media piece (video on Probability and Sampling) in this weekâ€™s Learning Resources. Reflect on the influence ofÂ Since a true census is seldom possible, most surveys collect data on only a portion or sample of the population.
Reference: Basic biostatistics: statistics for public health practice, (2nd ed.). Sudbury, Mass.: Jones and Bartlett Publishers. Custom Laureate Edition

" data-hasqtip="18" >sample sizewith respect to variability in a probability distribution. Then, consider the relationship between frequency and probability.

The Assignment: (1â€“2 pages)

Explain why increasing theSince a true census is seldom possible, most surveys collect data on only a portion or sample of the population.

Reference: Basic biostatistics: statistics for public health practice, (2nd ed.). Sudbury, Mass.: Jones and Bartlett Publishers. Custom Laureate Edition

" data-hasqtip="19" >sample sizedecreased the variability in the interactive media piece.Explain how frequency is used to inform probability and why this important. Be sure to include the relevance ofThe main &ldquo;output&rdquo; of hypothesis testing is the P-value. The P-value is the probability of the data or data that are more extreme assuming the null hypothesis is correct.

Reference: Basic biostatistics: statistics for public health practice, (2nd ed.). Sudbury, Mass.: Jones and Bartlett Publishers. Custom Laureate Edition

" data-hasqtip="20" >p-valuesas it concerns probability and the relationship withClick the link below to watch the video, Frequency Distributions.

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The Normal Distribution and Probability The Normal Distribution and Probability
Program Transcript
JENNIFER ANN MORROW: Welcome to the Normal Distribution Probability. My
name is Dr. Jennifer Ann Morrow. In today's demonstration, I will show you what
a Z-score is, what the properties of Z-scores are, what the formula is for a Zscore, and how to calculate a Z-score using both a formula as well as SPSS. I
will also go over with you probability in a normal distribution. I will show you how
to use the unit normal table. And lastly, I will show you how to determine the
probability of a score in your distribution. OK, let's get started.
A Z-score is a standardized score for a distribution of scores for a variable. Zscores identify and describe the exact location of every score in a distribution. Zscores serve two useful purposes. First, a Z-score tells us the exact location of
the original score, or raw score, within a distribution. Second, when you
transform a distribution into Z-scores, you can now compare that distribution of
scores to other distributions as standardized scores. There are a few things I
want to go over with you before we move on to an example.
First, the sign of a Z-score is very important. Whether or not a Z-score is positive
or negative tells you where it is located in the distribution. A Z-score that has a
value that is positive is located above the mean in a distribution. In other words, it
represents a score that is larger than the mean. A Z-score that has a value that is
negative is below the mean distribution. In other words, it represents a score that
is smaller than the mean.
Next, it's also important to know that the mean of a Z-score distribution is always
going to be 0, and the standard deviation is going to be 1. This is why it's known
as a standardized distribution. Also, the numerical value of the Z-score tells us
the distance between the score and the mean in terms of standard deviations.
For example, a Z-score that has a value of positive 1 tells us that the value is one
standard deviation above the mean. A Z-score with a value of negative 1 tells us
that the score is one standard deviation below the mean. And lastly, the shape of
a Z-score distribution is always going to be the same as the shape of the original
or raw score distribution. So when you change the raw scores into Z-scores, it
will not change the shape of that distribution.
As you can see, a Z-score distribution is very useful. Now, let me show you a
picture of a normal distribution. This is a picture of a standardized, or Z-score,
normal distribution. In a standardized normal distribution, the same number of
scores appear below the mean as they do appear above the mean. In other
words, a standardized Z-score distribution is symmetrical. You can fold the
distribution in half, and half the scores appear below the mean, and half the
scores appear above the mean. As you can see looking at this distribution, the xaxis, or horizontal axis, contains the values for the Z-scores. Â©2014 Laureate Education, Inc. 1 The Normal Distribution and Probability The value of 0 Z-score is the value of the mean of the distribution. It's directly in
the middle of the distribution. The value of positive 1 is one standard deviation
above the mean. 2 is two standard deviations above the mean. And 3 is three
standard deviations above the mean.
On the opposite side of the distribution, negative 1 is one standard deviation
below the mean. Negative 2 is two standard deviations below the mean. And
negative 3 is three standard deviations below the mean. As you can see, looking
at this picture, 68% of the scores fall between negative 1 and positive 1 Z-scores.
Approximately 34% of the scores fall between 0 and positive 1, and 34% of the
scores fall between 0 and negative 1. 95% of the scores in a standardized Zscore distribution fall between the values of negative 2, Z-score, and positive 2,
Z-score. And 99.7%, or nearly all of the scores in the distribution, fall between
negative 3, Z-score, and positive 3, Z-score.
Now you know what a standardized Z-score distribution looks like, let's move on
to calculate a Z-score using the formula. The formula for a Z-score is as follows.
z, or the value of a Z-score, equals x minus x bar divided by sd, where x is the
raw score in that distribution, x bar is the mean of that distribution, and sd is the
standard deviation for that distribution.
Now, let me go over an example use the Z-score formula. So again, z equals x
minus x bar divided by sd. If I know that my raw score is 85 and the mean for my
distribution equals 100 and my standard deviation for my distribution of scores
equals 15, how do I find the Z-score for the corresponding value of 85 in my
distribution? Well, now let's plug-in this information into the Z-score formula.
So Z equals 85 minus 100 divided by 15, negative 1.00. So the Z-score for the
value of 85 in my distribution is negative 1. Another way of putting that is for the
value of 85, it occurs one standard deviation below the mean in this distribution
of scores. But what about if I have a Z-score, and I want to find out what the
corresponding raw score value is for a distribution? You can easily do that as
well if you know this formula. Let me give you an example.
So again, remember, the Z-score formula is z equals x minus x bar over the
standard deviation. Now, I know my value for my Z-score. I have a z value of
2.00. Say it's positive 2.00. I know the mean of my distribution is 50. And I know
that the standard deviation of my distribution is 10. If I know this information, how
do I calculate and find the raw score?
So here, I have 2.00 equals x minus 50 over 10. I cross-multiply. 2 times 10
equals x minus 50. Then, 2 times 10 plus 50. I just moved the 50 to the other
side of the equation, equals x, or 20 plus 50 equals x, which equals 70. So if I
know that my distribution has a mean of 50 and a standard deviation of 10 and
I'm given the Z-score value of positive 2.00, I know that the raw score is a 70. Â©2014 Laureate Education, Inc. 2 The Normal Distribution and Probability Now, let me show you how to calculate Z-scores in SPSS. First, open SPSS and
find the data set that you want to use. Once you've found your data set, click on it
To calculate Z-scores for a variable in your data set, you're going to click on
Analyze, Descriptive Statistics, Descriptives, and now your descriptives dialog
box is open in SPSS. Now, choose a variable that you want to calculate the Zscores for for each of the raw scores in that variable. I'm going to choose selfesteem first semester. So click on self-esteem first semester, and then click on
the right arrow to move that over to the box on the right under the label Variables.
Next, we're going to click on the box on the far left. Save Standardized Values as
variables. And what SPSS is going to do is create a new column of data in our
data set that contains the Z-scores for each of the raw scores for the variable
self-esteem first semester. You can click on Paste to paste the syntax to your
syntax file and run your analyses from there, or you could just click on OK. Let's
just click on OK.
SPSS will create an output window and has the value self-esteem first semester.
It creates a descriptives table that shows you have 149 participants with a
minimum value of 1 and a maximum value of 3.90 and that the mean of that
variable, self-esteem first semester, is 1.7235, or running that up, 1.72. It has a
standard deviation of 0.74334, or rounding that up, 0.74.
But what you're most interested in are the Z-scores for this particular variable. To
see those, you actually need to go to the data window. So we can click on
Window. And then, we can click on to view the data. Now, we'll now open up our
data window. We're going to scroll all the way to the right. So what it's going to
do is SPSS has created that variable at the very end of our data set. As you can
see here, SPSS is creating a new column of data in our data set as the Z-score
for each of the values in self-esteem first semester. Right here, z, self-esteem.
An easy way for you to view both the raw score and the standardized score for a
particular variable is to create a case summaries table. Let me show you how to
do that. Let's click on Analyze. Reports. Case Summaries. And your Summarized
Cases dialog box window will appear on your screen. And now, you need to
choose your raw variable and your new Z-score of that variable to create your
table. In the variable box on the left, click on self-esteem first semester. And
then, click on the right arrow button to move it to the Variables box on the right.
Then, in your left variable box, scroll to the bottom. And then, click on your new
Z-score variable. And then, click on the right arrow button, and move that one
over also to the variable box on the right. Now, over here on the left, make sure
the display cases and show only valid cases buttons are checked. So we want
display cases and show only valid cases are checked. And limit cases to first and
show cases numbers are unchecked. So I want to uncheck limit cases to first and Â©2014 Laureate Education, Inc. 3 The Normal Distribution and Probability show case numbers are unchecked. Next, we want to click on the Statistics
buttons on our far right.
In the Show Statistics box, click on Number of Cases. And then, click on the left
arrow button to move that back over to the Statistics box on the left. We want the
Show Statistics box here on the right to be completely empty. Now, click on
Continue. And now we're ready to create our table. You can either click on the
Paste button to paste the syntax in your syntax file and run it from there, or you
can click on the OK button. So let's click on the OK button.
As you can see, SPSS has created a table, the second table in your output, that
is a case summary table that has each raw score and its corresponding Z-score.
So in this case, a raw score of 1.10 for the variable self-esteem first semester
has a Z-score value of negative 0.83877. So a score of 1.10, and you know that
by looking at the negative value, you know it falls below the mean of that
particular variable because of where it is in terms of the Z-score. So let's move
on.
We can take any raw score in a normal distribution and determine its probability.
First, we have to change the raw score to a Z-score. And then, we use something
called a unit normal table to find a specific proportion for the Z-score. So we can
answer two questions using something called a unit normal table. We can find
out the proportion of scores that fall below a particular Z-score, and we can find
the proportion of scores that fall above a particular Z-score.
Let me show you what a unit normal table looks like. Here's just a portion of the
unit normal table. You can find an example of a unit normal table in the back of
your statistics textbook in the appendices. Looking at your unit normal table, in
column A is the Z-score value. In column B is the proportion in the body of the
distribution. And this is always a larger portion in your distribution. In Column C,
it's the proportion in the tail of the distribution, or the smaller part of the
distribution. In some unit normal tables, there's also a column D, which is the
portion between the mean and the Z-score. But you can easily find that
information by just taking the value that occurs in Column B, the proportion in the
body, and subtracting 0.500. And you'll be able to get that value that would
appear in Column D.
Another thing that you should be aware of is that if you add the value in B to the
value in C, you're always going to get 1.00. Let me show you an example to
determine the probability of scores using the unit normal table. To determine the
probability of a score, you first transform the raw scores into Z-scores. Second,
you use the unit normal table to look up the proportions that correspond to the Zscore values. Let me show you an example.
Again, the formula for Z-score is z equals x minus x bar over the standard
deviation. If I have a Z-score of positive 0.25 and I want to find out, what is the Â©2014 Laureate Education, Inc. 4 The Normal Distribution and Probability proportion of scores below positive 0.25 and what is the proportion of scores
above positive 0.25, the first thing you should do is draw the distribution. And that
So if I have a Z-score of positive 0.25, I know that is going to be somewhere right
here, that it occurs above the mean of that distribution. So the larger portion of
my distribution, or the scores that occur below 0.25, is going to be the value that
occurs in the column B once I look at my unit normal table. For the question,
what's the proportion of scores above positive 0.25, that is a smaller portion of
my distribution. That is going to correspond to the value in Column C of my unit
normal table.
When I look at my unit normal table, I find out that for a Z-score of positive 0.25,
my value of B equals 0.5987 and my value of C equals 0.4013. So for the
question, what is the proportion of scores that occur below positive 0.25 Zscores, so the proportion that is less than positive 0.25, that is a larger portion of
my distribution, that equals B. So that equals 59.87%. So 59.87% of my scores in
my distribution appear below the Z-score of positive 0.25. And the proportion of
scores that occur above positive 0.25 equals the proportion of C, the smaller end
of my distribution, the tail end. And that equals 40.13%, so 40.13% of the scores
in my distribution occur above positive 0.25.
So when you are trying to look at and find the proportion of scores that occur
below or above a positive Z-score, the value in Column B is always going to be
the proportion below low a positive Z-score, and the value in Column C is always
going to be above a positive Z-score. But what about if you want to find the
proportion for a negative Z-score. You can easily do this as well. Let me show
you an example.
Again, Z-score formula is x minus x bar over the standard deviation. And I have a
Z-score value of negative 0.30. If I want to find out the values that occur below,
the proportion that occurs below the mean, below the Z-score of negative 0.30
and above the Z-score of negative 0.30, again, draw the distribution, and that will
So I know that the Z-score value of negative 0.30 is going to fall somewhere
around here. So values that occur below negative 0.3 is going to be represented
in Column C of my unit normal distribution. And the larger portion that occurs
above 0.3 is going to be represented in Column B of my unit normal table. So I
go to my unit normal table, and I look up the proportion that occurs under column
B and under column C for the Z-score of negative 0.30.
Now, in your unit normal table, there's only positive values. But you can look up
the Z-score of 0.30, positive 0.30, and be able to get the same information. So
you go to your unit normal table. And you look under a Z-score of 0.30. And you
find out that the value of B equals 0.6179, and the value of C equals 0.3821. So Â©2014 Laureate Education, Inc. 5 The Normal Distribution and Probability you know that for the values that occur above negative 0.30 is 61.79%. So
61.79% of the values in my distribution are above negative 0.30. And to find the
portion of values less than negative 0.30, that equals C, or 38.21%. So here,
38.21% of the values in my distribution occur below the Z-score value of negative
0.30.
We have now come to the end of our demonstration. Practice calculating Zscores on your own to gain more experience. The Normal Distribution and Probability
SPSS tutorials demonstrated by:
Jennifer Ann Morrow, PhD
Associate Professor of Evaluation, Statistics, and Measurement
University of Tennessee
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Los Angeles, CA
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Narrator Tracks Music Library
Stevens Point, WI
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Chesterton, IN
Studio Cutz Music Library
Carrollton, TX Â©2014 Laureate Education, Inc. 6

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