7.4 The Mean

1. Population Versus Sample2. Statistic Versus Parameter3. Mean (Average) of a Sample4. Mean (Average) of a Population5. Expected Value6. Expected Value of Binomial Trial

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Population Versus Sample

A 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.

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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.

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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.

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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 .rr

f f fx x x x

n n n

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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 f

N

2

1 21 .rr

f f fx x xN N N

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Note: Greek letters are used for parameters.

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?

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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

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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.

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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

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Example Expected Value Five coins are tossed and the number of

heads observed. Find the expected value.

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Example Expected Value

5132 32

10 1032 32

5 132 32

0 1

2 3

4 5

2.5

X

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Expected Value of Binomial Trial

X is a binomial random variable with parameters n and p, then

E(X) = np.

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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

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Fair Game

The expected value of a completely fair game is zero.

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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.

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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.

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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.

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