Ch7 - YSU

Chapter 7

Sampling Distributions

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Chapter Outline     

Selecting A Sample Point Estimation Introduction to Sampling Distributions Sampling Distribution of x Sampling Distribution of p

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Introduction  The reason we select a sample is to collect data to answer a research question about a population.  To directly study the population is probably too costly and too time-consuming, while to study a portion of it, i.e. a sample is much more manageable.  The sample results provide only estimates of the values of the population characteristics.  With proper sampling methods, the sample results can provide ‘good’ estimates of the population characteristics. 3

Sampling from A Finite Population  A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.  Replacing each sampled element before selecting subsequent elements is called sampling with replacement. Sampling without replacement is the procedure used most often.  In large sampling projects, computer-generated random numbers are often used to automate the sample selection process. 4

Sampling from An Infinite Population  It’s impossible to obtain a list of all elements in an infinite population.  Populations are often generated by an ongoing process where there is no upper limit on the number of units that can be generated.  A random sample must be selected with the following conditions satisfied:  Each element selected comes from the population of interest.  Each element is selected independently.

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Point Estimation  Point estimation is a form of statistical inference.  In point estimation we use the data from the sample to compute a value of a sample statistic the serves as an estimate of a population parameter. Sample Statistics

Point Estimates

Population Parameters

Sample mean: X

Population mean: 

Sample variance: S 2

Population variance: 

Sample proportion: P

Population proportion: P

2

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Point Estimation  Example: Checking Accounts A local small bank has a total of 600 checking accounts. The average daily balance of all the checking accounts is $310 with a standard deviation of $66. The proportion of accounts with a daily balance of no less than $500 is 30%. To find point estimates for the population, a random sample of 121 checking accounts at the bank are chosen, which shows an average daily balance of $306. The standard deviation of the sample is $61, and the sample proportion of accounts with a daily balance of no less than $500 is 27%. The following table summarizes the point estimates from the sample of 121 checking accounts. 7

Summary of Point Estimates of A Simple Random Sample of 121 Checking Accounts Population Parameter

Parameter Value

Point Estimator

Point Estimate

 = Population mean

$310

x = Sample mean

$306

s = Sample std. deviation for account balance

$61

account balance

 = Population std.

$66

deviation for account balance

p = Population proportion of account balance no less than $500

.3

account balance

p = Sample pro-

.27

portion of account balance no less than $500 8

Sampling Distribution of x The sampling distribution of x is the probability distribution of all possible values of the sample mean x .  Expected Value of x E( x ) =  where:  = the population mean When the expected value of the point estimator equals the population parameter, we say the point estimator is unbiased. 9

Sampling Distribution of x  Standard Deviation of x Finite Population

Infinite Population

N n  x  ( ) N 1 n

x 

 n

x = the standard deviation of x

 = the standard deviation of the population n = the sample size N = the population size 10

Sampling Distribution of x  Standard Deviation of x Finite Population

N n  x  ( ) N 1 n

Infinite Population

x 

 n

• A finite population is treated as being infinite if n/N < .05. • ( N  n) / ( N  1) is the finite population correction factor. •  x is referred to as the standard error of the mean. 11

Central Limit Theorem When the population from which we are selecting a random sample does not have a normal distribution, the central limit theorem is helpful in identifying the shape of the sampling distribution of x . CENTRAL LIMIT THEOREM In selecting random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal distribution as the sample size becomes large. 12

Sampling Distribution of x  When the population follows a normal distribution, the sampling distribution of x is normally distributed for any sample size.  According to the Central Limit Theorem, when the sample size is large enough (at least 30 in most cases), the sampling distribution of x can be approximated by a normal distribution.  The point of studying the sampling distribution of x is to better estimate the population mean  . 13

Sampling Distribution of x  Process of Statistical Inference Population with mean =?

The value of x is used to make inferences about the value of .

A simple random sample of n elements is selected from the population.

The sample data provide a value for the sample mean x .

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Sampling Distribution of x  Example: Checking Accounts 

Suppose the population of checking accounts follows a normal distribution, a random sample of 121 checking accounts should also follow a normal distribution with the mean account balance of $310 and the standard error of 6.

x 

E x   $310

66 121

6

x

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Sampling Distribution of x  Example: Checking Accounts 

What is the probability that a simple random sample of 121 checking accounts will provide an point estimate of the population mean account balance within +/- $5 of the actual population mean? (i.e. between $305 and $315)

x $305

$315

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Sampling Distribution of x  Example: Checking Accounts 

Since x follows a normal distribution, we can first calculate the z values of both cutoff points as follows:

z

x  E x 

305  310   0.83 6

x

315  310   0.83

x

6

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Sampling Distribution of x  Example: Checking Accounts 

Next, we check the cumulative probabilities for the standard normal distribution table for areas corresponding to the z values. Recall that the numbers in the table represent the areas under the standard normal curve to the LEFT of z values.

z

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

.

.

.

.

.

.

.

.

.

.

.

.5

.6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

.6

.7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

.7

.7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

.8

.7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

.9

.8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

.

.

.

.

.

.

.

.

.

.

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Sampling Distribution of x  Example: Checking Accounts 

The area under the standard normal curve to the left of 0.83

Area = .7967

0

0.83

z 19

Sampling Distribution of x  Example: Checking Accounts 

The area under the standard normal curve to the left of -0.83

Area = 1-0.7967 = 0.2033 -0.83 0

z 20

Sampling Distribution of x  Example: Checking Accounts 

The area we are looking for is the area under the normal curve between the cutoff points. Therefore, the probability is calculated as

P(-.83 < z < .83) = P(z < .83) - P(z < -.83) = .7967 - .2033 = .5934 The probability that the sample mean account balance will be between $305 and $315 is:

P(305 <

x < 315) = .5934 21

Sampling Distribution of x  Example: Checking Accounts

x  6 Area = .5934

$305 $310 $315

x 22

Relationship Between the Sample Size and the Sampling Distribution of x  As the sample size increases, the standard error  x becomes smaller.

x 

 n

 With a smaller standard error, the values of x have less variability and tend to be closer to the population mean.

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Relationship Between the Sample Size and the Sampling Distribution of x  Example: Checking Accounts

With n = 225,  x  4.4

With n=121 , x  6

E x   $310

x

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Sampling Distribution of p The sampling distribution of p is the probability distribution of all possible values of the sample proportion p .  Expected Value of p

E ( p)  p where: p = the population proportion

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Sampling Distribution of p  Standard Deviation of p Finite Population

N  n p(1  p) p  N 1 n

Infinite Population

p(1  p) p  n

•  p is referred to as the standard error of the proportion.

• ( N  n) / ( N  1) is the finite population correction factor. When n/N is  5%, the correction factor is close to 1. 26

Sampling Distribution of p The sampling distribution of p can be approximated by a normal distribution whenever the sample size is large enough to satisfy the two conditions: np > 5

and

n(1 – p) > 5

. . . because when these conditions are satisfied, the probability distribution of x in the sample proportion, p = x/n, can be approximated by normal distribution (and because n is a constant).

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Sampling Distribution of p Making Inferences about a Population Proportion Population with proportion p=?

The value of p is used to make inferences about the value of p.

A simple random sample of n elements is selected from the population.

The sample data provide a value for the sample proportion p. 28

Sampling Distribution of p  Example: Checking Accounts

 In the example, n = 121 and p = .3, the sampling distribution of p can be approximated as a normal distribution because: np = 121(.30) = 36.3 > 5 and n(1-p) = 121(.70) = 84.7 > 5  The standard error of p is:

p 

600 121  600 1

.31  .3  0.037 121 29

Sampling Distribution of p  Example: Checking Accounts

 p  0.037

E p   0.3

p

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