Statistika Ekonomi I : Distribusi Probabilitas Normal Dan Binomial
Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use...
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![Page 1: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/1.jpg)
Review• Normal Distributions
– Draw a picture.– Convert to standard normal (if necessary)– Use the binomial tables to look up the value.
– In the case of a reverse look up we may have to now solve for x.
![Page 2: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/2.jpg)
Estimator and Point Estimate
An estimator is a “sample statistic” (such as the sample mean, or sample standard deviation) used to approximate a population parameter.
A Point Estimate is a single value or point used to approximate a population parameter. A point estimator may be biased or unbiased.
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Central Limit Theorem
Take ANY random variable X and compute and for this variable. If samples of size n are randomly selected from the population, then:
1) For large n, the distribution of the sample means, will be approximately a normal distribution,
2) The mean of the sample means will be the population mean and
3) The standard deviation of the sample means will be
n
x
![Page 4: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/4.jpg)
![Page 5: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/5.jpg)
The Sampling Distribution of X and the Central Limit Theorem
Assume a population with = 80, = 6. If a sample of 36 is taken from this population, what is the probability that the sample mean is larger than 82?
Sketch the curve of x and identify area of interest
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The Sampling Distribution of X and the Central Limit Theorem
Convert 82 to z value
First, calculate the standard deviation of the sampling distribution
Then calculate the z value
Use the tables to find probability of interest
82 802
1x
x
xz
61.0
36x n
82 ( 2) .5 .4772 .0228P x P z
![Page 7: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/7.jpg)
Practice Problems
• #6.34 page 310
• #6.41 page 311
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Example
Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213.
What can you say about the average balance?
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Example
Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213.
What can you say about the average balance of all VISA accounts?
ANS: That it is approximately $213. (This is a point estimate.
![Page 10: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/10.jpg)
Example
Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213.
If in addition you know the standard deviation for all accounts is $112, what can you say about the average balance of all VISA accounts?
![Page 11: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/11.jpg)
Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213.
If in addition you know the standard deviation for all accounts is $112, what can you say about the average balance of all VISA accounts?
You can create an interval (called a confidence interval) that you can be 95% sure the mean is in.
![Page 12: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/12.jpg)
Confidence Interval
A confidence interval (or interval estimate) is a range of values that estimates the true value of the population parameter.
This is associated with a degree of confidence, which is a measure the probability that a randomly selected confidence interval encloses the population parameter.
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Confidence LevelThe confidence level is equal to 1- , and is split between the two tails of the distribution
Usually the confidence level is:
90% (meaning )
95% (meaning ) OR
99% (meaning ) 10.05.01.
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Confidence Intervals
The Confidence Interval is expressed as:
E is called the margin of error.
For samples of size > 30,
nzxzx
Ex
x
22
n
szx 2
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Example
Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213. If in addition you know the standard deviation for all accounts is $112, what can you say about the average balance of all VISA accounts? Build a 95% confidence interval.
![Page 16: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/16.jpg)
Example
Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213. If in addition you know the standard deviation for all accounts is $112, what can you say about the average balance of all VISA accounts? Build a 95% confidence interval.
95.21213
![Page 17: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/17.jpg)
Example
Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213. If in addition you know the standard deviation for all accounts is $112, what can you say about the average balance of all VISA accounts? Build a 90% confidence interval.
100
112645.1213
![Page 18: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/18.jpg)
Example
Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213. If in addition you know the standard deviation for all accounts is $112, what can you say about the average balance of all VISA accounts? Build a 90% confidence interval.
42.18213100
112645.1213
![Page 19: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/19.jpg)
Example
Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213. If in addition you know the standard deviation for all accounts is $112, what can you say about the average balance of all VISA accounts? Build a 90% confidence interval.
100
112575.2213
![Page 20: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/20.jpg)
Example
Suppose you took a sample of 100 accounts from Visa and found the mean balance was $213. If in addition you know the standard deviation for all accounts is $112, what can you say about the average balance of all VISA accounts? Build a 90% confidence interval.
84.28213100
112575.2213
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Age of STFX Students
Find the 90% confidence interval for the mean.
0.10 50n ,6.2 ,4.21 sx
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Age of STFX Students
Find the 90% confidence interval for the mean.
22.0 20.850
6.24.21 50
6.24.21
0.10 50n ,6.2 ,4.21
05.005.0
22
zz
nzx
nzx
sx
![Page 23: Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.](https://reader036.fdocuments.net/reader036/viewer/2022062409/5697bff91a28abf838cbf810/html5/thumbnails/23.jpg)
Sample Size
The sample size needed to estimate so as to be (1-)*100 % confident that the sample mean does not differ from more than E is:
…round up
2
2
E
zn
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Overview
• Confidence Intervals
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Practice Problems
• #7.11 page 329
• #7.19 page 331
• #7.21 page 331
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26
Homework
• Review Chapter 7.1, 7.2
• Read Chapters 7.3-7.5