The document discusses properties of normal distributions and the standard normal distribution. It provides examples of finding probabilities and values associated with normal distributions. The key points are:
- Normal distributions are continuous and bell-shaped. The mean, median and mode are equal.
- The standard normal distribution has a mean of 0 and standard deviation of 1.
- Probabilities under the normal curve can be found using z-scores and the standard normal table.
- Values like z-scores can be determined by finding the corresponding cumulative area in the standard normal table.
Normal Distribution – Introduction and PropertiesSundar B N
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Normal Distribution – Introduction and PropertiesSundar B N
In this video you can see Normal Distribution – Introduction and Properties.
Watch the video on above ppt
https://www.youtube.com/watch?v=ocTXHLWsec8&list=PLBWPV_4DjPFO6RjpbyYXSaZHiakMaeM9D&index=4
Subscribe to Vision Academy
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The Normal Distribution is a symmetrical probability distribution where most results are located in the middle and few are spread on both sides. It has the shape of a bell and can entirely be described by its mean and standard deviation.
Topic: Variance
Student Name: Sonia Khan
Class: B.Ed. 2.5
Project Name: “Young Teachers' Professional Development (TPD)"
"Project Founder: Prof. Dr. Amjad Ali Arain
Faculty of Education, University of Sindh, Pakistan
The Normal Distribution is a symmetrical probability distribution where most results are located in the middle and few are spread on both sides. It has the shape of a bell and can entirely be described by its mean and standard deviation.
Topic: Variance
Student Name: Sonia Khan
Class: B.Ed. 2.5
Project Name: “Young Teachers' Professional Development (TPD)"
"Project Founder: Prof. Dr. Amjad Ali Arain
Faculty of Education, University of Sindh, Pakistan
TOPIC OUTLINE: 1. The Normal Curve
a. Definition/Description
b. Area Under Normal Curve
2. Standard Scores
a. Z-Scores
b. T-Scores
c. Other Standard Scores
Karl Friedrich Gauss:
one of the scientist that developed the concept of normal curve.
Normal Curve
is a continuous probability distribution in statistics
Karl Pearson:
first to refer to the curve as “Normal Curve”
Asymptotic:
approaching the x-axis but never touches it
Symmetric:
made up of exactly similar parts facing each other
STANDARD SCORES
-is a raw score that has been converted from one scale to another scale.
Z-scores
called a zero plus or minus one scale
Scores can be positive and negative
T-Scores
a none of the scores is negative. It can be called a 50 plus or minus ten scale. ( 50 mean set and 10 SD set )
Stanine: Standard Nine
(STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.
Our concepts of heart disease are based on the enormous reservoir of physiologic and anatomic knowledge derived from the past 70 years' of experience in the cardiac catheterization laboratory.
As Andre Cournand remarked in his Nobel lecture of December 11, 1956, the cardiac catheter was the key in the lock.
By turning this key, Cournand and his colleagues led us into a new era in the understanding of normal and disordered cardiac function in huma
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Chapter 6: Normal Probability Distribution
6.1: The Standard Normal Distribution
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
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
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PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
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The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
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- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
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Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
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Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
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Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
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3. Properties of Normal Distributions A continuous random variable has an infinite number of possible values that can be represented by an interval on the number line. The probability distribution of a continuous random variable is called a continuous probability distribution. Hours spent studying in a day 0 6 3 9 15 12 18 24 21 The time spent studying can be any number between 0 and 24.
4. Properties of Normal Distributions The most important probability distribution in statistics is the normal distribution . A normal distribution is a continuous probability distribution for a random variable, x. The graph of a normal distribution is called the normal curve . Normal curve x
5.
6. Properties of Normal Distributions μ 3 σ μ + σ μ 2 σ μ σ μ μ + 2 σ μ + 3 σ x Inflection points Total area = 1 If x is a continuous random variable having a normal distribution with mean μ and standard deviation σ , you can graph a normal curve with the equation
7. Means and Standard Deviations A normal distribution can have any mean and any positive standard deviation. Mean: μ = 3.5 Standard deviation: σ 1.3 Mean: μ = 6 Standard deviation: σ 1.9 The mean gives the location of the line of symmetry. The standard deviation describes the spread of the data. Inflection points Inflection points 3 6 1 5 4 2 x 3 6 1 5 4 2 9 7 11 10 8 x
8.
9. Interpreting Graphs Example : The heights of fully grown magnolia bushes are normally distributed. The curve represents the distribution. What is the mean height of a fully grown magnolia bush? Estimate the standard deviation. The heights of the magnolia bushes are normally distributed with a mean height of about 8 feet and a standard deviation of about 0.7 feet. The inflection points are one standard deviation away from the mean. σ 0.7 μ = 8 6 8 7 9 10 Height (in feet) x
10. The Standard Normal Distribution The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Any value can be transformed into a z - score by using the formula The horizontal scale corresponds to z - scores. 3 1 2 1 0 2 3 z
11. The Standard Normal Distribution If each data value of a normally distributed random variable x is transformed into a z - score, the result will be the standard normal distribution. After the formula is used to transform an x - value into a z - score, the Standard Normal Table in Appendix B is used to find the cumulative area under the curve. The area that falls in the interval under the nonstandard normal curve (the x - values) is the same as the area under the standard normal curve (within the corresponding z - boundaries). 3 1 2 1 0 2 3 z
12.
13. The Standard Normal Table Example : Find the cumulative area that corresponds to a z - score of 2.71. Find the area by finding 2.7 in the left hand column, and then moving across the row to the column under 0.01. The area to the left of z = 2.71 is 0.9966. Appendix B: Standard Normal Table .6141 .6103 .6064 .6026 .5987 .5948 .5910 .5871 .5832 .5793 0.2 .5753 .5714 .5675 .5636 .5596 .5557 .5517 .5478 .5438 .5398 0.1 .5359 .5319 .5279 .5239 .5199 .5160 .5120 .5080 .5040 .5000 0.0 .09 .08 .07 .06 .05 .04 .03 .02 .01 .00 z .9981 .9980 .9979 .9979 .9978 .9977 .9977 .9976 .9975 .9974 2.8 .9974 .9973 .9972 .9971 .9970 .9969 .9968 .9967 .9966 .9965 2.7 .9964 .9963 .9962 .9961 .9960 .9959 .9957 .9956 .9955 .9953 2.6
14. The Standard Normal Table Example : Find the cumulative area that corresponds to a z - score of 0.25. Find the area by finding 0.2 in the left hand column, and then moving across the row to the column under 0.05. The area to the left of z = 0.25 is 0.4013 Appendix B: Standard Normal Table .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003 3.3 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002 3.4 .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 z .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4724 .4681 .4641 0.0 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .4247 0.1 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .3859 0.2 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3483 0.3
15.
16.
17.
18. Guidelines for Finding Areas Example : Find the area under the standard normal curve to the left of z = 2.33. From the Standard Normal Table, the area is equal to 0.0099. Always draw the curve! 2.33 0 z
19. Guidelines for Finding Areas Example : Find the area under the standard normal curve to the right of z = 0.94. From the Standard Normal Table, the area is equal to 0.1736. Always draw the curve! 0.8264 1 0.8264 = 0.1736 0.94 0 z
20. Guidelines for Finding Areas Example : Find the area under the standard normal curve between z = 1.98 and z = 1.07. From the Standard Normal Table, the area is equal to 0.8338. Always draw the curve! 0.8577 0.0239 = 0.8338 0.8577 0.0239 1.07 0 z 1.98
21. § 5.2 Normal Distributions: Finding Probabilities
22. Probability and Normal Distributions If a random variable, x , is normally distributed, you can find the probability that x will fall in a given interval by calculating the area under the normal curve for that interval. P ( x < 15) μ = 10 σ = 5 15 μ = 10 x
23. Probability and Normal Distributions Same area P ( x < 15) = P ( z < 1) = Shaded area under the curve = 0.8413 15 μ = 10 P ( x < 15) μ = 10 σ = 5 Normal Distribution x 1 μ = 0 μ = 0 σ = 1 Standard Normal Distribution z P ( z < 1)
24. Probability and Normal Distributions Example : The average on a statistics test was 78 with a standard deviation of 8. If the test scores are normally distributed, find the probability that a student receives a test score less than 90. P ( x < 90) = P ( z < 1.5) = 0.9332 The probability that a student receives a test score less than 90 is 0.9332. 1.5 μ = 0 z ? 90 μ = 78 P ( x < 90) μ = 78 σ = 8 x
25. Probability and Normal Distributions Example : The average on a statistics test was 78 with a standard deviation of 8. If the test scores are normally distributed, find the probability that a student receives a test score greater than than 85. P ( x > 85) = P ( z > 0.88) = 1 P ( z < 0.88) = 1 0.8106 = 0.1894 The probability that a student receives a test score greater than 85 is 0.1894. 0.88 μ = 0 z ? 85 μ = 78 P ( x > 85) μ = 78 σ = 8 x
26. Probability and Normal Distributions Example : The average on a statistics test was 78 with a standard deviation of 8. If the test scores are normally distributed, find the probability that a student receives a test score between 60 and 80. P (60 < x < 80) = P ( 2.25 < z < 0.25) = P ( z < 0.25) P ( z < 2.25) The probability that a student receives a test score between 60 and 80 is 0.5865. 0.25 2.25 = 0.5987 0.0122 = 0.5865 μ = 0 z ? ? 60 80 μ = 78 P (60 < x < 80) μ = 78 σ = 8 x
28. Finding z-Scores Example : Find the z - score that corresponds to a cumulative area of 0.9973. Find the z - score by locating 0.9973 in the body of the Standard Normal Table. The values at the beginning of the corresponding row and at the top of the column give the z - score. The z - score is 2.78. Appendix B: Standard Normal Table 2.7 .08 .6141 .6103 .6064 .6026 .5987 .5948 .5910 .5871 .5832 .5793 0.2 .5753 .5714 .5675 .5636 .5596 .5557 .5517 .5478 .5438 .5398 0.1 .5359 .5319 .5279 .5239 .5199 .5160 .5120 .5080 .5040 .5000 0.0 .09 .08 .07 .06 .05 .04 .03 .02 .01 .00 z .9981 .9980 .9979 .9979 .9978 .9977 .9977 .9976 .9975 .9974 2.8 .9974 .9973 .9972 .9971 .9970 .9969 .9968 .9967 .9966 .9965 2.7 .9964 .9963 .9962 .9961 .9960 .9959 .9957 .9956 .9955 .9953 2.6
29. Finding z-Scores Example : Find the z - score that corresponds to a cumulative area of 0.4170. Find the z - score by locating 0.4170 in the body of the Standard Normal Table. Use the value closest to 0.4170. Appendix B: Standard Normal Table The z - score is 0.21. 0.2 .01 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003 0.2 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002 3.4 .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 z .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4724 .4681 .4641 0.0 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .4247 0.1 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .3859 0.2 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3483 0.3 Use the closest area.
30. Finding a z-Score Given a Percentile Example : Find the z - score that corresponds to P 75 . The z - score that corresponds to P 75 is the same z - score that corresponds to an area of 0.75. The z - score is 0.67. 0.67 ? μ = 0 z Area = 0.75
31. Transforming a z-Score to an x-Score To transform a standard z - score to a data value, x, in a given population, use the formula Example : The monthly electric bills in a city are normally distributed with a mean of $120 and a standard deviation of $16. Find the x - value corresponding to a z - score of 1.60. We can conclude that an electric bill of $145.60 is 1.6 standard deviations above the mean.
32. Finding a Specific Data Value Example : The weights of bags of chips for a vending machine are normally distributed with a mean of 1.25 ounces and a standard deviation of 0.1 ounce. Bags that have weights in the lower 8% are too light and will not work in the machine. What is the least a bag of chips can weigh and still work in the machine? The least a bag can weigh and still work in the machine is 1.11 ounces. P ( z < ?) = 0.08 P ( z < 1.41 ) = 0.08 1.11 ? 0 z 8% 1.41 1.25 x ?
33. § 5.4 Sampling Distributions and the Central Limit Theorem
34. Sampling Distributions Sample A sampling distribution is the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. Sample Sample Sample Sample Sample Sample Sample Sample Sample Population
35. Sampling Distributions If the sample statistic is the sample mean, then the distribution is the sampling distribution of sample means . Sample 1 Sample 4 Sample 3 Sample 6 The sampling distribution consists of the values of the sample means, Sample 2 Sample 5
36.
37.
38.
39.
40.
41.
42. The Central Limit Theorem the sample means will have a normal distribution . If a sample of size n 30 is taken from a population with any type of distribution that has a mean = and standard deviation = , x x
43. The Central Limit Theorem If the population itself is normally distributed , with mean = and standard deviation = , the sample means will have a normal distribution for any sample size n . x
44. The Central Limit Theorem In either case, the sampling distribution of sample means has a mean equal to the population mean. Mean of the sample means Standard deviation of the sample means The sampling distribution of sample means has a standard deviation equal to the population standard deviation divided by the square root of n. This is also called the standard error of the mean .
45. The Mean and Standard Error Example : The heights of fully grown magnolia bushes have a mean height of 8 feet and a standard deviation of 0.7 feet. 38 bushes are randomly selected from the population, and the mean of each sample is determined. Find the mean and standard error of the mean of the sampling distribution. Mean Standard deviation (standard error) Continued.
46. Interpreting the Central Limit Theorem Example continued : The heights of fully grown magnolia bushes have a mean height of 8 feet and a standard deviation of 0.7 feet. 38 bushes are randomly selected from the population, and the mean of each sample is determined. From the Central Limit Theorem, because the sample size is greater than 30, the sampling distribution can be approximated by the normal distribution. The mean of the sampling distribution is 8 feet ,and the standard error of the sampling distribution is 0.11 feet.
47. Finding Probabilities Example : The heights of fully grown magnolia bushes have a mean height of 8 feet and a standard deviation of 0.7 feet. 38 bushes are randomly selected from the population, and the mean of each sample is determined. Find the probability that the mean height of the 38 bushes is less than 7.8 feet. The mean of the sampling distribution is 8 feet, and the standard error of the sampling distribution is 0.11 feet. Continued. 7.8
48. Finding Probabilities 1.82 Example continued : Find the probability that the mean height of the 38 bushes is less than 7.8 feet. The probability that the mean height of the 38 bushes is less than 7.8 feet is 0.0344. = 0.0344 P ( < 7.8) = P ( z < ____ ) ? 7.8 P ( < 7.8)
49. Probability and Normal Distributions Example : The average on a statistics test was 78 with a standard deviation of 8. If the test scores are normally distributed, find the probability that the mean score of 25 randomly selected students is between 75 and 79. 0.63 1.88 Continued. 0 z ? ? P (75 < < 79) 75 79 78
50. Probability and Normal Distributions Example continued : Approximately 70.56% of the 25 students will have a mean score between 75 and 79. = 0.7357 0.0301 = 0.7056 0.63 1.88 0 z ? ? P (75 < < 79) P (75 < < 79) = P ( 1.88 < z < 0.63) = P ( z < 0.63) P ( z < 1.88) 75 79 78
51. Probabilities of x and x Example : The population mean salary for auto mechanics is = $34,000 with a standard deviation of = $2,500. Find the probability that the mean salary for a randomly selected sample of 50 mechanics is greater than $35,000. 2.83 = P ( z > 2.83) = 1 P ( z < 2.83) = 1 0.9977 = 0.0023 The probability that the mean salary for a randomly selected sample of 50 mechanics is greater than $35,000 is 0.0023. 0 z ? 35000 34000 P ( > 35000)
52. Probabilities of x and x Example : The population mean salary for auto mechanics is = $34,000 with a standard deviation of = $2,500. Find the probability that the salary for one randomly selected mechanic is greater than $35,000. 0.4 = P ( z > 0.4) = 1 P ( z < 0.4) = 1 0.6554 = 0.3446 The probability that the salary for one mechanic is greater than $35,000 is 0.3446. (Notice that the Central Limit Theorem does not apply.) 0 z ? 35000 34000 P ( x > 35000)
53. Probabilities of x and x Example : The probability that the salary for one randomly selected mechanic is greater than $35,000 is 0.3446. In a group of 50 mechanics, approximately how many would have a salary greater than $35,000? P ( x > 35000) = 0.3446 This also means that 34.46% of mechanics have a salary greater than $35,000. You would expect about 17 mechanics out of the group of 50 to have a salary greater than $35,000. 34.46% of 50 = 0.3446 50 = 17.23
54. § 5.5 Normal Approximations to Binomial Distributions
55. Normal Approximation The normal distribution is used to approximate the binomial distribution when it would be impractical to use the binomial distribution to find a probability. Normal Approximation to a Binomial Distribution If np 5 and nq 5, then the binomial random variable x is approximately normally distributed with mean and standard deviation
56.
57. Correction for Continuity The binomial distribution is discrete and can be represented by a probability histogram. This is called the correction for continuity . When using the continuous normal distribution to approximate a binomial distribution, move 0.5 unit to the left and right of the midpoint to include all possible x - values in the interval. To calculate exact binomial probabilities, the binomial formula is used for each value of x and the results are added. Exact binomial probability c P ( x = c ) P (c 0.5 < x < c + 0.5) Normal approximation c c + 0.5 c 0.5
58.
59.
60. Approximating a Binomial Probability Example : Thirty - one percent of the seniors in a certain high school plan to attend college. If 50 students are randomly selected, find the probability that less than 14 students plan to attend college. np = (50)(0.31) = 15.5 nq = (50)(0.69) = 34.5 P ( x < 13.5) = P ( z < 0.61) = 0.2709 The probability that less than 14 plan to attend college is 0.2079. The variable x is approximately normally distributed with = np = 15.5 and Correction for continuity 10 15 x 20 = 15.5 13.5
61. Approximating a Binomial Probability Example : A survey reports that forty - eight percent of US citizens own computers. 45 citizens are randomly selected and asked whether he or she owns a computer. What is the probability that exactly 10 say yes? np = (45)(0.48) = 12 nq = (45)(0.52) = 23.4 P (9.5 < x < 10.5) = 0.0997 The probability that exactly 10 US citizens own a computer is 0.0997. = P ( 0.75 < z 0.45) Correction for continuity 5 10 x 15 = 12 9.5 10.5