MATH 1280-01 Introduction to Statistics - AY2023-
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Discussion 7
by Anson Xuan (Instructor) - Wednesday, 5 April 2023, 6:26 PM
Discussion Assignment
In the discussion forum, you are expected to participate often and engage in deep levels of discourse. You are required to post an
initial response to the question/issue presented in the Forum by Sunday evening and then respond to at least 3 of your
classmates’ initial posts. You should also respond to anyone who has responded to you.
Based on what you have studied in this unit, reflect and answer the following questions:
What are the main characteristics of the normal distribution?
Why is it advantageous to use the Z scores?
For a standard normal distribution, about 68% of values falls within one standard deviation of the mean; about 95%
falls within two standard deviations; and about 99.7% falls within three standard deviations. Suppose the distribution
of math scores in the SAT follow a normal distribution with a mean µ=700 and a standard deviation σ=150.
1. Calculate the Z-score for an SAT Math score of 600. Interpret it in your own word.
2. Find the interval [a, b] that captures 68% of the SAT Math score in this example.
3. Find the interval [a, b] that captures 95% of the SAT Math score in this example.
4. Find the interval [a, b] that captures 99.7% of the SAT Math score in this example.
Your Discussion should be a minimum of 200 words in length and not more than 500 words. Please include a word count.
Following the APA standard, use references and in-text citations for the textbook and any other sources.
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Discussion 7
by Izabella Areizaga - Thursday, 18 May 2023, 4:17 PM
Normal distributions contain symmetry, meaning that the data is equally distributed on both sides of the mean. They also
have a bell-shaped curve with the highest frequency of values occurring at the mean and decreasing as you get further
from the mean. Central tendency is another part of normal distributions where the mean, median, and mode are located
at the center. Last but noy least, they follow the empirical rule where approximately 68% of the values fall within one
standard deviation of the mean, 95% fall within two , and 99.7% fall within three.
Utilizing Z-scores is beneficial because they enable us to standardize and compare values from various normal
distributions. The number of standard deviations a value has from the mean is shown by a Z-score. We can compare
values and their relative positions in the distribution by converting them into Z-scores.
In order to find the Z-score for SAT Math score of 600, we use the formula: Z = (x - μ) / σ where x is the value, μ is the
mean, and σ is the standard deviation.
Z = (600 - 700) / 150 Z = -1/3
This means that the SAT Math score of 600 has a Z-score of -1/3. 600 score is one-third of a standard deviation BELOW
the mean.
To find the interval a, b that captures 68% of the SAT Math scores, we need to use empirical rule. Since 68% falls within
one standard deviation of the mean, we can find the interval by adding and subtracting one standard deviation from the
mean. a = μ - σ a = 700 - 150 a = 550, b = μ + σ b = 700 + 150 b = 850
So we find that the interval (a, b) that captures 68% of the SAT Math scores is (550, 850).
When looking for the interval that captures 95%, we consider two standard deviations from the mean. a = μ - 2σ a = 700
- 2(150) a = 400, b = μ + 2σ b = 700 + 2(150) b = 1000
The interval that captures 95% of the SAT Math scores is (400, 1000).
The one that captures 99.7% of the SAT Math scores, we consider three standard deviations from the mean. a= μ - 3σ a =
700 - 3(150) a = 250, b = μ + 3σ b = 700 + 3(150) b = 1150
Is (250, 1150).
References
Frost, J. (2022). Z-score: Definition, Formula, and Uses. Statistics by Jim. https://statisticsbyjim.com/basics/z-score/
Beyer, A. (2021, August 1). Chapter 6: z-scores and the Standard Normal Distribution.
Pressbooks. https://open.maricopa.edu/psy230mm/chapter/chapter-6-z-scores/
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Re: Discussion 7
by Anson Xuan (Instructor) - Thursday, 18 May 2023, 11:14 PM
,Yes, you are correct. Normal distributions exhibit symmetry, with the data being equally distributed on both sides of
the mean. The bell-shaped curve shows that the highest frequency of values occurs at the mean and decreases as
you move away from the mean.
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Re: Discussion 7
by Alejandro Montoya - Friday, 19 May 2023, 6:38 PM
Hello Izabella.
Your calculations are correct and your process is easy to follow. It is true that the normal distribution model is a bell-
shaped curve that is symmetrical about the mean. This is well explained in your own words. Keep it up.
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Re: Discussion 7
by Mudassar Hussain - Saturday, 20 May 2023, 7:56 AM
Hi Izabella
You did a great job, your assigment is mind blowing. I agreed to utilizing Z-scores is beneficial because they
enable us to standardize and compare values from various normal distributions. The number of standard
deviations a value has from the mean is shown by a Z-score. We can compare values and their relative positions
in the distribution by converting them into Z-scores.
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Re: Discussion 7
by Farhiya Hassan - Tuesday, 23 May 2023, 5:56 PM
Great work on explaining
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Re: Discussion 7
by Jadon Louis - Tuesday, 23 May 2023, 11:53 PM
I must commend you on a job well done this week Izabella! Your presentation of the normal distribution was spot on
and I agree with your statement on the Z-scores being efficient because they enable us to standardize and compare
values from various normal distributions. Your calculations are not incorrect and the structure of your post resembles
that of proper study material. Thank you and good luck with your studies.
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Re: Discussion 7
by Nabila Khalily - Wednesday, 24 May 2023, 12:02 PM
, Hello Izabella.
Your calculations are correct and your process is easy to follow. It is true that the normal distribution model is a bell-
shaped curve that is symmetrical about the mean. This is well explained in your own words. Keep it up.
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Re: Discussion 7
by Mahesh Pandit - Wednesday, 24 May 2023, 11:41 PM
Your explanation of the properties of normal distributions, such as symmetry, bell-shaped curve, and central
tendency, is accurate and clear. It provides a good understanding of the basic characteristics of normal distributions.
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Re: Discussion 7
by Alejandro Montoya - Thursday, 18 May 2023, 7:32 PM
Based on what you have studied in this unit, reflect, and answer the following questions:
What are the main characteristics of the normal distribution?
The Normal Distribution, also referred to as the Gaussian Distribution, is a widely used continuous distribution model
real-world random values. As observed by Illowsky et al. (2022), This type of distribution is very popular as it is
represented in various distinct sectors, including: psychology, business, economics, science, nursing, and of course
mathematics; its application is vastly used and understood as a representation of data values (pg.365). The normal
distribution model is characterized by some key components (You can see a model below in the graph):
Symmetry: The model is a bell-shaped curve that rises and falls along the horizontal axis demonstrating how the values
are symmetrical about the mean(μ).
Mean, Median, Mode: In this type of distribution, the mean, media, and mode are all the same due to the symmetry about
the mean.
Boundless: Because the mean can shift left or right, this means that the model is not bounded by extrema values and thus
it has infinite number of normal probability distributions.
Empirical Rule: This rule represents how the data is distributed along the curve and it follows a set of specific dilemmas;
for example, 68% of the data falls within one standard deviation of the mean, 95% fall within. Two standard deviations,
and 99.7% fall within three standard deviations.
Why is it advantageous to use the Z scores?
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