7.4 The Mean
description
Transcript of 7.4 The Mean
![Page 1: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/1.jpg)
7.4 The Mean1. Population Versus Sample2. Statistic Versus Parameter3. Mean (Average) of a Sample4. Mean (Average) of a Population5. Expected Value6. Expected Value of Binomial Trial
1
![Page 2: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/2.jpg)
Population Versus SampleA population is a set of all elements about
which information is desired. A sample is a subset of a population that is analyzed in an attempt to estimate certain properties of the entire population.
2
![Page 3: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/3.jpg)
Example Population Versus Sample
A clothing manufacturer wants to know what style of jeans teens between 13 and 16 will buy. To help answer this question, 200 teens between 13 and 16 were surveyed.
The population is all teens between 13 and 16.
The sample is the 200 teens between 13 and 16 surveyed.
3
![Page 4: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/4.jpg)
Statistic Versus Parameter A numerical descriptive measurement
made on a sample is called a statistic. Such a measurement made on a population is called a parameter of the population.
Since we cannot usually have access to entire populations, we rely on our experimental results to obtain statistics, and we attempt to use the statistics to estimate the parameters of the population.
4
![Page 5: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/5.jpg)
Mean (Average) of a Sample
Let an experiment have as outcomes the numbers x1, x2, …, xr with frequencies f1, f2,…, fr, respectively, so that f1 + f2 +…+ fr = n. Then the sample mean equals
or
21 1 2 ,r rx f x f x fx
n
2
1 21 .rrf f fx x x xn n n
5
![Page 6: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/6.jpg)
Mean (Average) of a Population
If the population has x1, x2,…, xr with frequencies f1, f2,…, fr, respectively. Then the population mean equals
or
21 1 2 ,r rx f x f x fN
2
1 21 .rrf f fx x xN N N
6
Note: Greek letters are used for parameters.
![Page 7: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/7.jpg)
Example Mean An ecologist observes
the life expectancy of a certain species of deer held in captivity. The table shows the data observed on a population of 1000 deer. What is the mean life expectancy of this population?
7
![Page 8: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/8.jpg)
Example Mean (2) The relative
frequencies are given in the table.
1(0) 2(.06) 3 .18
4 .25 5 .20
6 .12 7 .05
8 .12 9 .02
4.87
8
![Page 9: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/9.jpg)
Expected Value The expected value of the random
variable X which can take on the values x1, x2,…,xN with
Pr(X = x1) = p1, Pr(X = x2) = p2,…, Pr(X = xN) = pN
is E(X) = x1p1 + x2p2+ …+ xNpN.
9
![Page 10: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/10.jpg)
Expected Value (2)
The expected value of the random variable X is also called the mean of the probability distribution of X and is also designated by
The expected value of a random variable is the center of the probability distribution in the sense that it is the balance point of the histogram.
.X
10
![Page 11: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/11.jpg)
Example Expected Value Five coins are tossed and the number of
heads observed. Find the expected value.
11
![Page 12: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/12.jpg)
Example Expected Value
5132 32
10 1032 32
5 132 32
0 1
2 3
4 5
2.5
X
12
![Page 13: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/13.jpg)
Expected Value of Binomial Trial
X is a binomial random variable with parameters n and p, then
E(X) = np.
13
![Page 14: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/14.jpg)
Example Expected Value Binomial Trial
Five coins are tossed and the number of heads observed. Find the expected value.
A "success" is a head and p = .5. The number of trials is n = 5. E(X) = np = 5(.5) = 2.5
14
![Page 15: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/15.jpg)
Fair Game The expected value of a completely fair
game is zero.
15
![Page 16: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/16.jpg)
Example Fair Game Two people play a dice game. A single die
is thrown. If the outcome is 1 or 2, then A pays B $2. If the outcome is 3, 4, 5, or 6, then B pays A $4. What are the long-run expected winnings for A?
X represents the payoff to A. Therefore, X is either -2 or 4.
16
![Page 17: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/17.jpg)
Example Fair Game (2) Pr(X = -2) = 2/6 = 1/3 Pr(X = 4) = 4/6 = 2/3 E(X) = -2(1/3) + 4(2/3) = 2 On average, A should expect to win $2
per play. If A paid B $4 on 1 and 2 but B paid A $2
on 3, 4, 5, and 6, then E(X) = 0 and the game would be fair.
17
![Page 18: 7.4 The Mean](https://reader036.fdocuments.us/reader036/viewer/2022062305/568160e3550346895dd0138f/html5/thumbnails/18.jpg)
Summary Section 7.4 The sample mean of a sample of n
numbers is the sum of the numbers divided by n.
The expected value of a random variable is the sum of the products of each outcome and its probability.
18