![Page 1: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/1.jpg)
Proportions
![Page 2: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/2.jpg)
A proportion is the fraction of individuals having a particular attribute.
![Page 3: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/3.jpg)
Example: 2092 adult passengers on the
Titanic;
654 survived
Proportion of survivors = 654/2092 0.3
![Page 4: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/4.jpg)
Probability that two out of three randomly chosen passengers survived the Titanic
![Page 5: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/5.jpg)
Binomial distribution
The binomial distribution describes the probability of a given number of "successes" from a fixed number of independent trials.
![Page 6: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/6.jpg)
n trials; p probability of success
![Page 7: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/7.jpg)
6! = 6 * 5 * 4 * 3 * 2 *1 = 720
0!=11!=1
n! = n * n-1 * n-2 *... 3 * 2 * 1
![Page 8: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/8.jpg)
Probability that two out of three randomly chosen passengers survived the Titanic
![Page 9: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/9.jpg)
Probability that two out of three randomly chosen passengers survived the Titanic
Number of ways to get 2 survivors out of 3 passengers
Probability of 2 survivors
Probability of 1 death
![Page 10: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/10.jpg)
Example: Paradise flycatchersA population of paradise flycatchers has 80% brown males and 20% white. Your field assistant captures 5 male flycatchers at random.
What is the chance that 3 of those are brown and 2 are white?
![Page 11: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/11.jpg)
p = 0.8 n = 5 X = 3
Pr[3] =
![Page 12: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/12.jpg)
Try at home:
What is the probability that 3 or more are brown?
![Page 13: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/13.jpg)
Try at home:
What is the probability that 3 or more are brown?
![Page 14: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/14.jpg)
Properties of the binomial distribution:Number of successes
![Page 15: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/15.jpg)
Binomial distribution for p = 0.8, n = 5
![Page 16: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/16.jpg)
Proportion of successes in a sample
The hat (^) shows that this is an estimate of p.
![Page 17: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/17.jpg)
Properties of sample proportions
Mean: p
Variance:
![Page 18: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/18.jpg)
Estimating a proportion
![Page 19: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/19.jpg)
Standard error of the estimate of a proportion
![Page 20: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/20.jpg)
A larger sample has a lower standard error
![Page 21: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/21.jpg)
The law of large numbers
The greater the sample size, the closer an estimate of a proportion is likely to be to its true value.
Sample size
![Page 22: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/22.jpg)
95% confidence interval for a proportion
This is the Agresti-Coull confidence interval
![Page 23: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/23.jpg)
Example: The daughters of radiologists
30 out of 87 offspring of radiologists are males, and the rest female. What is the best estimate of the proportion of sons among radiologists?
![Page 24: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/24.jpg)
Example: The daughters of radiologists
30 out of 87 offspring of radiologists are males, and the rest female. What is the best estimate of the proportion of sons among radiologists?
![Page 25: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/25.jpg)
Example: The daughters of radiologists
30 out of 87 offspring of radiologists are males, and the rest female. What is the best estimate of the proportion of sons among radiologists? What is the 95% confidence interval for this estimate?
= 30/87, or 0.345
![Page 26: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/26.jpg)
Example: The daughters of radiologists
30 out of 87 offspring of radiologists are males, and the rest female. What is the best estimate of the proportion of sons among radiologists? What is the 95% confidence interval for this estimate?
![Page 27: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/27.jpg)
Example: The daughters of radiologists
30 out of 87 offspring of radiologists are males, and the rest female. What is the best estimate of the proportion of sons among radiologists? What is the 95% confidence interval for this estimate?
![Page 28: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/28.jpg)
Example: The daughters of radiologists
30 out of 87 offspring of radiologists are males, and the rest female. What is the best estimate of the proportion of sons among radiologists? What is the 95% confidence interval for this estimate?
![Page 29: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/29.jpg)
Hypothesis testing on proportions
The binomial test
![Page 30: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/30.jpg)
Binomial test
The binomial test uses data to test whether a population proportion p matches a null expectation for the proportion.
H0: The relative frequency of successes in the population is p0 .
HA: The relative frequency of successes in the population is not p0 .
![Page 31: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/31.jpg)
Example
An example: Imagine a student takes a multiple choice test before starting a statistics class. Each of the 10 questions on the test have 5 possible answers, only one of which is correct. This student gets 4 answers right. Can we deduce from this that this student knows anything at all about statistics?
![Page 32: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/32.jpg)
Hypotheses
H0: Student got correct answers randomly.
HA: Student got more answers correct than random.
![Page 33: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/33.jpg)
Hypotheses
H0: Student got correct answers randomly.
H0: p0 = 0.2.
![Page 34: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/34.jpg)
N =10, p0 =0.2
![Page 35: Proportions. A proportion is the fraction of individuals having a particular attribute](https://reader030.vdocuments.net/reader030/viewer/2022032702/56649f495503460f94c6b116/html5/thumbnails/35.jpg)
P = 0.12
This is greater than the value of 0.05, so we would not reject the null hypothesis.
It is plausible that the student had four answers correct just by guessing randomly.