Normal Approximation of the Binomial Distribution
description
Transcript of Normal Approximation of the Binomial Distribution
![Page 1: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/1.jpg)
Normal Approximation of
the Binomial Distribution
![Page 2: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/2.jpg)
What is the Probability of getting exactly 30 tails if a coin is tossed
50 times?X = tails, x = 30n = 50, p = 0.5
P(X = x) = 5030 (0.5)30(1 – 0.5)50 - 30
= 0.042
![Page 3: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/3.jpg)
Find the probability that tails will occur less than 30 times….
The magnitude of the calculation motivates us to determine an easier way…
![Page 4: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/4.jpg)
Notice:The most frequent outcome of
flipping a coin should be 25 tails, then 24/26, then 23/27 and so on….
This structure has already been modeled and studied as a normal distribution!
![Page 5: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/5.jpg)
![Page 6: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/6.jpg)
Therefore
Under certain conditions, we may use the Normal Model to approximate probabilities from a Binomial Distribution!!
![Page 7: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/7.jpg)
Conditions that must be met
1. np > 52. nq > 5
![Page 8: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/8.jpg)
When we will use the Normal distribution to approximate the
Binomial distribution, we have to create a z score
![Page 9: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/9.jpg)
z = x - x
= np(1 – p)
x = E(x) = np
x = given value, but it must be corrected by 0.5 either way (if it is discrete…)
s
s
![Page 10: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/10.jpg)
Ex: If you toss a coin 50 times, estimate the probability that you will get tails less than 30 times.
To approximate, first we will check to see of the conditions
are met
![Page 11: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/11.jpg)
1. Is np > 5 ?
50(0.5) > 5 ?25 > 5 YES!
2. Is nq > 5?
50(0.5) > 5 ?25 > 5 YES!
If these conditions are met, that means there were enough trials so a comparable distribution was created…
![Page 12: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/12.jpg)
Now we need to determine the z score
z = x - xs
![Page 13: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/13.jpg)
Ex: If you toss a coin 50 times, estimate the probability that you will get tails less than 30 times.
Success: tailsn = 50p = 0.5E = x = 50(0.5)
= 25
![Page 14: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/14.jpg)
= 50(0.5)(1 – 0.5)
s = np(1 – p)
= 3.54
![Page 15: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/15.jpg)
Less than 30 tails..
x < 30Imagine the 30 bin
3029.5 30.5
![Page 16: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/16.jpg)
z = x - x
= 29.5 - 253.54
= 1.27P(X < 29.5) = P(z < 1.27)
= 0.8980
s
![Page 17: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/17.jpg)
A bank found that 24% of it’s loans become delinquent. If 200
loans are made, find the probability that at least 60 are
delinquent.
![Page 18: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/18.jpg)
We may analyze this situation as a binomial model because:
1. A loan is either paid back or not
2. The loans are all independent
![Page 19: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/19.jpg)
Check to see if we can approximate…
1. np = (200)(0.24)= 48 (which is greater than 5, so
yes)2. nq = (200)(0.76)= 152 (which is greater than 5, so
yes)
![Page 20: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/20.jpg)
E(X) = x = np= (200)(0.24)= 48
s = (npq)1/2
=[(200)(0.24)(0.76)]1/2
= 6.04
X = 60, then adjust to 59.5
![Page 21: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/21.jpg)
z = 59.5 – 48 6.04
z = 1.90 (97.13% from the chart)
Therefore, the number of delinquent loans is
100% - 97.13% = 2.87%
Sketch this to confirm!
![Page 22: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/22.jpg)
If we were to actually calculate the probability directly, it would come out to 3.07%
![Page 23: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/23.jpg)
Page 449
1,2[odd]3,5,6,8,10
![Page 24: Normal Approximation of the Binomial Distribution](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815a93550346895dc80b78/html5/thumbnails/24.jpg)
Note: Discrete to Continuouspg 306