Here are the steps to solve this problem:
a) Find the z-score corresponding to 115 mm Hg: (115 - 85)/13 = 2.31
The proportion that is NOT severely hypertensive is 1 - P(Z >= 2.31) = 1 - 0.0103 = 0.9897
b) Find the z-score corresponding to 90 mm Hg: (90 - 85)/13 = 0.3846
The proportion that will be asked to consult a physician is P(Z >= 0.3846) = 0.6507
c) Find the z-scores corresponding to the mildly hypertensive range:
(90 - 85)/13 = 0.3846
(
Chapter 4 part3- Means and Variances of Random Variablesnszakir
Statistics, study of probability, The Mean of a Random Variable, The Variance of a Random Variable, Rules for Means and Variances, The Law of Large Numbers,
Chapter 5 part1- The Sampling Distribution of a Sample Meannszakir
Mathematics, Statistics, Population Distribution vs. Sampling Distribution, The Mean and Standard Deviation of the Sample Mean, Sampling Distribution of a Sample Mean, Central Limit Theorem
Chapter 4 part3- Means and Variances of Random Variablesnszakir
Statistics, study of probability, The Mean of a Random Variable, The Variance of a Random Variable, Rules for Means and Variances, The Law of Large Numbers,
Chapter 5 part1- The Sampling Distribution of a Sample Meannszakir
Mathematics, Statistics, Population Distribution vs. Sampling Distribution, The Mean and Standard Deviation of the Sample Mean, Sampling Distribution of a Sample Mean, Central Limit Theorem
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Chapter 6: Normal Probability Distribution
6.1: The Standard Normal Distribution
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Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
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Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
When modeling a system, encountering missing data is common.
What shall a modeler do in the case of unknown or missing information?
When dealing with missing data, it is critical to make correct assumptions to ensure that the system is accurate.
One must common strategy for handling such situations is calculate the average of available data for the similar existing systems (i.e., creating sampling data).
Use this average as a reasonable estimate for the missing value.
The Normal Distribution:
There are different distributions namely Normal, Skewed, and Binomial etc.
Objectives:
Normal distribution its properties its use in biostatistics
Transformation to standard normal distribution
Calculation of probabilities from standard normal distribution using Z table.
Normal distribution:
- Certain data, when graphed as a histogram (data on the horizontal axis, frequency on the vertical axis), creates a bell-shaped curve known as a normal curve, or normal distribution.
- Two parameters define the normal distribution, the mean (µ) and the standard deviation (σ).
Properties of the Normal Distribution:
Normal distributions are symmetrical with a single central peak at the mean (average) of the data.
The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the mean.
Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean.
-The mean, the median, and the mode fall in the same place. In a normal distribution the mean = the median = the mode.
- The spread of a normal distribution is controlled by the
standard deviation.
In all normal distributions the range ±3σ includes nearly
all cases (99%).
Uni modal:
One mode
Symmetrical:
Left and right halves are mirror images
Bell-shaped:
With maximum height at the mean, median, mode
Continuous:
There is a value of Y for every value of X
Asymptotic:
The farther the curve goes from the mean, the closer it gets to the X axis but it never touches it (or goes to 0).
The total area under a normal distribution curve is equal to 1.00, or 100%.
Using Normal distribution for finding probability:
While finding out the probability of any particular observation we find out the area under the curve which is covered by that particular observation. Which is always 0-1.
Transforming normal distribution to standard normal distribution:
Given the mean and standard deviation of a normal distribution the probability of occurrence can be worked out for any value.
But these would differ from one distribution to another because of differences in the numerical value of the means and standard deviations.
To get out of this problem it is necessary to find a common unit of measurement into which any score could be converted so that one table will do for all normal distributions.
This common unit is the standard normal distribution or Z
score and the table used for this is called Z table.
- A z score always reflects the number of standard deviations above or below the mean a particular score or value is.
where
X is a score from the original normal distribution,
μ is the mean of the original normal distribution, and
σ is the standard deviation of original normal distribution.
Steps for calculating probability using the Z-
score:
-Sketch a bell-shaped curve,
- Shade the area (which represents the probability)
-Use the Z-score formula to calculate Z-value(s)
-Look up Z-values in table
Lecture 3 Facial cosmetic surgery
Maxillofacial Surgery
Dental Students Fifth Year second semester
Al Azhar University Gaza Palestine
Dr. Lama El Banna
https://twitter.com/lama_k_banna
Lecture 1 Facial cosmetic surgery
Maxillofacial Surgery
Dental Students Fifth Year second semester
Al Azhar University Gaza Palestine
Dr. Lama El Banna
https://twitter.com/lama_k_banna
Facial neuropathology Maxillofacial SurgeryLama K Banna
Lecture 4 facial neuropathology
Maxillofacial Surgery
Dental Students Fifth Year second semester
Al Azhar University Gaza Palestine
Dr. Lama El Banna
https://twitter.com/lama_k_banna
Lecture 2 Facial cosmetic surgery
Maxillofacial Surgery
Dental Students Fifth Year second semester
Al Azhar University Gaza Palestine
Dr. Lama El Banna
https://twitter.com/lama_k_banna
Lecture 12 general considerations in treatment of tmdLama K Banna
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Lecture Name 12 general considerations in the treatment of TMJ
Al Azhar University Gaza Palestine
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Maxillofacial Surgery
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Lecture 7 correction of dentofacial deformities Part 2Lama K Banna
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Lecture 8 management of patients with orofacial cleftsLama K Banna
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Lecture 5 Diagnosis and management of salivary gland disorders Part 2Lama K Banna
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Empowering the Data Analytics Ecosystem: A Laser Focus on Value
The data analytics ecosystem thrives when every component functions at its peak, unlocking the true potential of data. Here's a laser focus on key areas for an empowered ecosystem:
1. Democratize Access, Not Data:
Granular Access Controls: Provide users with self-service tools tailored to their specific needs, preventing data overload and misuse.
Data Catalogs: Implement robust data catalogs for easy discovery and understanding of available data sources.
2. Foster Collaboration with Clear Roles:
Data Mesh Architecture: Break down data silos by creating a distributed data ownership model with clear ownership and responsibilities.
Collaborative Workspaces: Utilize interactive platforms where data scientists, analysts, and domain experts can work seamlessly together.
3. Leverage Advanced Analytics Strategically:
AI-powered Automation: Automate repetitive tasks like data cleaning and feature engineering, freeing up data talent for higher-level analysis.
Right-Tool Selection: Strategically choose the most effective advanced analytics techniques (e.g., AI, ML) based on specific business problems.
4. Prioritize Data Quality with Automation:
Automated Data Validation: Implement automated data quality checks to identify and rectify errors at the source, minimizing downstream issues.
Data Lineage Tracking: Track the flow of data throughout the ecosystem, ensuring transparency and facilitating root cause analysis for errors.
5. Cultivate a Data-Driven Mindset:
Metrics-Driven Performance Management: Align KPIs and performance metrics with data-driven insights to ensure actionable decision making.
Data Storytelling Workshops: Equip stakeholders with the skills to translate complex data findings into compelling narratives that drive action.
Benefits of a Precise Ecosystem:
Sharpened Focus: Precise access and clear roles ensure everyone works with the most relevant data, maximizing efficiency.
Actionable Insights: Strategic analytics and automated quality checks lead to more reliable and actionable data insights.
Continuous Improvement: Data-driven performance management fosters a culture of learning and continuous improvement.
Sustainable Growth: Empowered by data, organizations can make informed decisions to drive sustainable growth and innovation.
By focusing on these precise actions, organizations can create an empowered data analytics ecosystem that delivers real value by driving data-driven decisions and maximizing the return on their data investment.
Explore our comprehensive data analysis project presentation on predicting product ad campaign performance. Learn how data-driven insights can optimize your marketing strategies and enhance campaign effectiveness. Perfect for professionals and students looking to understand the power of data analysis in advertising. for more details visit: https://bostoninstituteofanalytics.org/data-science-and-artificial-intelligence/
StarCompliance is a leading firm specializing in the recovery of stolen cryptocurrency. Our comprehensive services are designed to assist individuals and organizations in navigating the complex process of fraud reporting, investigation, and fund recovery. We combine cutting-edge technology with expert legal support to provide a robust solution for victims of crypto theft.
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Chatty Kathy - UNC Bootcamp Final Project Presentation - Final Version - 5.23...John Andrews
SlideShare Description for "Chatty Kathy - UNC Bootcamp Final Project Presentation"
Title: Chatty Kathy: Enhancing Physical Activity Among Older Adults
Description:
Discover how Chatty Kathy, an innovative project developed at the UNC Bootcamp, aims to tackle the challenge of low physical activity among older adults. Our AI-driven solution uses peer interaction to boost and sustain exercise levels, significantly improving health outcomes. This presentation covers our problem statement, the rationale behind Chatty Kathy, synthetic data and persona creation, model performance metrics, a visual demonstration of the project, and potential future developments. Join us for an insightful Q&A session to explore the potential of this groundbreaking project.
Project Team: Jay Requarth, Jana Avery, John Andrews, Dr. Dick Davis II, Nee Buntoum, Nam Yeongjin & Mat Nicholas
Adjusting primitives for graph : SHORT REPORT / NOTESSubhajit Sahu
Graph algorithms, like PageRank Compressed Sparse Row (CSR) is an adjacency-list based graph representation that is
Multiply with different modes (map)
1. Performance of sequential execution based vs OpenMP based vector multiply.
2. Comparing various launch configs for CUDA based vector multiply.
Sum with different storage types (reduce)
1. Performance of vector element sum using float vs bfloat16 as the storage type.
Sum with different modes (reduce)
1. Performance of sequential execution based vs OpenMP based vector element sum.
2. Performance of memcpy vs in-place based CUDA based vector element sum.
3. Comparing various launch configs for CUDA based vector element sum (memcpy).
4. Comparing various launch configs for CUDA based vector element sum (in-place).
Sum with in-place strategies of CUDA mode (reduce)
1. Comparing various launch configs for CUDA based vector element sum (in-place).
2. Recall
Continuous probability distribution
A continuous probability distribution can assume an
infinite number of values within a given range – for
variables that take continuous values.
The distance students travel to class.
The time it takes an executive to drive to work.
The length of an afternoon nap.
The length of time of a particular phone call.
The amount of money spent on your last haircut.
4. Normal Probability Distribution
The normal probability distribution is the most
important distribution for describing a continuous
random variable.
It has been used in a wide variety of
applications:
Heights and weights of people
Test scores
Scientific measurements
Amounts of rainfall
It is widely used in statistical inference
5. Normal Probability Distribution
Normal Probability Density Function
where:
µ = mean
σ = standard deviation
π = 3.14159
e = 2.71828
22
2/)(
2
1
)( σµ
σπ
−−
= x
exf
This formula generates the density curve which gives
the shape of the normal distribution.
6. According to the Germans:
“That’s right on the money”
2
2
1
2
1
−
−
σ
µ
πσ
X
e
Herr
Gauss
The Normal distribution is also known as
the Gaussian distribution
7. “Bell shaped” and Symmetrical
about the mean µ.
Unimodal and its mode
occurs at x = µ.
Mean, median and
mode are equal
Interquartile range
equals 1.33 σ
Random variable has infinite
range from –∞ to ∞.
The total area under the
curve and above the horizontal axis is equal to 1.
Properties of the Normal
Distribution
µ
x
f(x)
Mean
Median
Mode
8. Most
observations in
the distribution
are close to the
mean, with
gradually fewer
observations
further away
Properties of the Normal
Distribution
µ
x
f(x)
Mean
Median
Mode
9. Properties of the Normal
Distribution
Empirical Rule
68.3 % of all observations
lie within 1 standard
deviation of the mean
95.4 % of all observations
lie within 2 standard
deviations of the mean
99.7 % of all observations
lie within 3 standard
deviations of the mean
µ-3σ µ-2σ µ-1σ µ µ+1σ µ+2σ µ+3σ
68.26%
95.45%
99.74%
55 70 14585 100 115 130
µ-3σ µ-2σ µ-1σ µ µ+1σ µ+2σ µ+3σ
55 70 14585 100 115 130
For a normal distribution, the following rules hold true;
These rules are commonly
known as the Empirical Rule
10. Women participating in a three-day experimental
diet regime have been demonstrated to have
normally distributed weight loss with mean 600 g
and a standard deviation 200 g.
a) What percentage of these women will have a
weight loss between 400 and 800 g?
b) What percentage of women will lose weight too
quickly on the diet (where too much weight is
defined as >1000g)?
Application of the Empirical Rule
13. σ2
2
2
µ x
σ1
2
1
µ µ1 < µ2
a. The normal distribution depends on the values of the
parameters µ, the population mean and σ2, the
population variance.
a. Two normal curves, which have the same standard
deviation but different means.
Properties of the Normal Distribution
14. σ2
2
xµ1 = µ2
σ1
2
σ2
2
σ1
2
<
b. Two normal curves with the same mean but
different standard deviations
Properties of the Normal Distribution
15. σ2
2
xµ1 < µ2
σ1
2
σ2
2
σ1
2
<
µ1
µ2
c. Two normal curves that have different means and
different standard deviations.
Properties of the Normal Distribution
16. Areas under the Normal Curve
The area under the curve bounded by the two ordinates x = x1
and x = x2 equals the probability that the random variable x
assumes a value between x = x1 and x = x2. Thus, for the normal
curve in the Figure below, the P(x1 < x < x2) is represented by the
area of the shaded region.
µx1 x2
17. The area under the curve between any two ordinates
depend upon the values of µ and σ and consequently, the
probability associated with distributions differ in mean and
standard deviation will be different for the two given values
of x.
Areas under the Normal Curve
I
II
x1 x2
P(x1 < x < x2) for different Normal curves
18. Many Normal Distributions
By varying the parameters σ and µ, we
obtain different normal distributions
There are an infinite number of normal distributions
19. It would be hopeless task to set up separate
tables of normal curve areas for every
conceivable value of µ and σ
Areas under the Normal Curve
20. Which Table to Use?
An infinite number of normal distributions means an
infinite number of tables to look up!
Yet we must use tables if we hope to avoid the
use of integral calculus
21. P(x1 < x < x2)
= ?
Finding Probabilities
Probability is the
area under the
curve!
x1 x2
X
22. Fortunately, all the observations of any normal
random variable x could be transformed to a new
set of observations of a normal random variable z
with mean zero and variance 1, by using the
transformation namely,
σ
µ−
=
X
Z
The Standard Normal Distribution
23. σ
µ z1
σ =1
0x1
z2x2
Any Normal Distribution Standard Normal Distribution
Areas will be equal.
zx
The Standard Normal Distribution
The new distribution is called Standard Normal
Distribution, with mean equal to 0 and it’s standard
deviation equal to 1.
24. By standardising any normally distributed random
variable, we can use just the table namely,
Areas Under the Normal Curve
Or
Areas of a Standard Normal Distribution
Such tables are usually found in the Appendix of any
statistics book.
Standard Normal Distribution
Tables
27. Entries give the probability that a standard
normally distributed random variable will assume
a value to the left of a given z value.
This means we have to sometimes perform minor
calculations to determine probabilities.
Use of the Normal Probability Table
28. Example
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
Find the probability that z is less than 1.74.
1. Locate a value of z equal to 1.7 in the left column.
2. Move across the row to the column under 0.04, where we
read 0.9591.
Therefore,
P(z < 1.74) = 0.9591.
z
29. 1. State the problem.
2. What is the appropriate probability
statement?
3. Draw a picture and shade required area
4. Convert to a standard normal distribution
5. Find the probability in the standard normal
table
Computing Normal Probabilities
30. Example: Suppose the number of a
particular type of bacteria in samples of
1-ml of drinking water tend to be
approximately normally distributed with µ
=85 and σ2
= 81.
What is the probability that a given 1-ml
sample will contain more than 100
bacteria?
31. σ
85 100
Find this area
σ = 9
We are to find the P(x> 100)
Sketch a curve
To find the P(x> 100), we need to evaluate the area
under the normal curve to the right of x = 100.
x
32. transform x = 100 to the corresponding z value
σ
µ−
= 1
1
x
z
Since µ = 85 and σ = 981 =
9
85100 −
=z = 1.67
33. 0 1.67
σ = 1
Find this area
z
Hence
P(x> 100) = P(z > 1.67)
Since the area to the right and the table only
gives the probability to the left of the
distribution we use
P(z > 1.67) = 1 − P(z < 1.67)
35. 0 1.67
σ = 1
Find this area
z
Hence
P(x> 100) = P(z > 1.67)
Since the area to the right and the table only gives the
probability to the left of the distribution we use
P(z > 1.67) = 1 − P(z < 1.67)
= 1 − 0.9525 = 0.0475.
So 1-ml sample will has 0.0475 probability to contain more
than 100 bacteria
36. Exercise
If the total cholesterol values for a certain target population
are approximately normally distributed with a mean of 200
(mg/100 mL) and a standard deviation of 20 (mg/100 mL),
What is the probability that a person picked at random from
this population will have a cholesterol value greater than 240
(mg/100 mL)?
Answer = 0.0228
37. Example 2
IQ scores (IQ is Intelligence Quotient) on the
Wechsler Adult Intelligence Scale are
approximately normally distributed with mean µ =
100 and σ = 15.
a. What is the proportion of persons having IQs
between 80 and 120?
38. Solution
Let x1 = 80 and x2 = 120
This question means that we are to find the probability
P(x1<IQ<x2) = P(80< IQ <120)
Sketch a normal curve like the one in the figure below.
Shade in the area desired.
σ =15
10080 120 x
Example 2 cont.
39. Find the z values corresponding to x1 = 80 and x2 = 120
Example 2 cont.
Since, P(x1<x<x2) =P (z1<z<z2)
Therefore, P(80<IQ<120) = P(−1.33<z<1.33)
40. The P(−1.33<z<1.33) is given by the area of the shaded region
in the figure. This area may be found by subtracting the area
to the left of the ordinate z = −1.33 from the entire area to the
left of z = 1.33.
Example 2 cont.
σ = 1
0-1.33 1.33 z
41. P(80<IQ<120) = P(−1.33<z<1.33)
= P(z<1.33) − P(z<−1.33)
= 0.9082 – 0.0918
= 0.8164.
Therefore the proportion of persons
having IQs between 80 and 120
is 0.8164, about 82%.
σ =1
0-1.33 1.33 z
42. Example 3
Suppose that the hemoglobin levels for healthy adult males
are approximately normally distributed with a mean of 16
and variance of 0.81. Find the probability that a randomly
chosen healthy adult male has a hemoglobin level less than
14.
Solution
We are to find P(x < 14)
Since µ = 16 and σ = = 0.9
We find Z = − 2.22
P(x < 14) = P(Z < − 2.22)
= 0.0132
Therefore, only about 1.3% of healthy adult males have
σ=0.81
σ= 1
16
0
14
-2.22
x
z
45. 1. State the problem
2. Draw a picture
3. Use table to find the probability closest to the one you
need
4. Read off the z-value
5. Unstandardise i.e. x = µ + zσ
Finding a Value (X) given a Probability
46. Example
Given a normal distribution with µ = 40 and σ = 6,
find the value of x that has
a. 45% of the area below it
b. 14% of the area above it.
Solution
In this problem we reverse the process and begin
with a known area or probability, find the z value,
and then determine x by rearranging the formula
to give
σ
µ−
=
x
z µσ += zx
47.
48. a. We require a z value that leaves an area of 0.45 to the left.
From the table we find P(z < -0.13) = 0.45 so that the desired
z value is -0.13.
σ=6
40 x
0.45
µσ += zx
= (-0.13 X 6 )+ 40 = 39.22
x =?
Example cont.
49. b. 14% of the area above it.
This time we require a z value that leaves 0.14 of the area to
the right and hence an area of 0.86 to the left.
From the table we find P(z < 1.08) = 0.86
so that the desired z value is 1.08 and
x = (6)(1.08) + 40
= 46.48
σ =6
x =?
0.86
40
µσ += zx
Example cont.
0.14
50.
51. Classification of arterial diastolic blood pressure (mm Hg) in
adults, 18 years and older
Consider that blood pressure readings are obtained from nearly
200,000 participants in a a large-scale community blood pressure
screening program, and that these measurements follow a normal
distribution. The mean is 85 mm Hg, with a standard deviation of 13
mm Hg.
Diastolic blood pressure (mm Hg) Blood Pressure Classification
Less than 85 Normal
85-89 High normal
90-104 Mild hypertension
105-114 Moderate hypertension
Greater or equal to 115 Severe hypertension
Exercise
52. a) What proportion of our sample will NOT be categorised
as severely hypertensive?
b) Suppose that we recommend that a physician be
consulted if an individual has an arterial diastolic blood
pressure equal or greater than 90 mm Hg. What
proportion of individuals in our screening program will be
asked to consult a physician?
c) What proportion of individuals have diastolic blood
pressure in the mildly hypertensive range?
d) What diastolic blood pressure will 75% of the
population be above?
Exercise
53. EXERCISE
(a) The distribution of serum levels of alpha tocopherol
(serum vitamin E) is approximately normal with mean
860 mg/dL and standard deviation 340mg/dL.
What serum level will 85% of the population be
below?
(b) Suppose a person is identified as having toxic
levels of alpha tocopherol if his or her serum level is
greater than 2000mg/dL. What percentage of people
will be so identified?
54. Under certain conditions, a binomial random
variable has a distribution that is approximately
normal.
Normal approximation of the
Binomial Distribution
55. If n, p, and q are such
that:
np and nq
are both greater than
5.
We use a normal distribution with
qpnandpn =σ=µ
Normal approximation of the
Binomial Distribution
56. Application of Normal Distribution
If 22% of all patients with high blood pressure
have side effects from a certain medication, and
100 patients are treated, find the probability that
at least 30 of them will have side effects.
Using the Binomial Probability Formula we would need
to compute:
P(30) + P(31) + ... + P(100) or 1 − P( x < 29)
57. Using the Normal Approximation to
the Binomial Distribution
Check if Is it appropriate to use the normal
distribution?
n p = 22
n q = 78
Both are greater than five.
58. Find the mean and standard deviation
µ = n p = 100(.22) = 22
and σ =
14.416.17
)78)(.22(.100
=
=
qpnandpn =σ=µ
Using the Normal Approximation to
the Binomial Distribution
59. σ
22 30
Find this area
To find the probability that at least 30 of
them will have side effects, find P( x ≥ 30)
σ = 4.14
Using the Normal Approximation to the
Binomial Distribution
60. Applying the Normal Distribution
z = 30 – 22 = 1.81
4.14
Find P( z ≥ 1.81) = 1 – P (z < 1.81) = 0.0351
0 1.81
The probability that at least 30 of the patients will have side
effects is 0.0351.
σσ = 1