1

Chapter 8 Goals1. Explain why a sample is the only feasible

way to learn about a population2. Describe methods to select a sample:

• Simple Random Sampling• Systematic Random Sampling• Stratified Random Sampling

3. Sampling Error4. Sampling Distribution Of The Sample

Mean5. Central Limit Theorem

2

So Far, & The Future…Chapter 2-4

Descriptive statistics about something that has already happened: Frequency distributions, charts, measures of central tendency,

dispersion

Chapter 5, 6, 7 Probability:

Probability Rules Probability Distributions

Probability distributions encompass all possible outcomes of an experiment and the probability associated with each outcome

We use probability distributions to evaluate something that might occur in the future

Discrete Probability Distributions : Binomial

Continuous Probability Distributions: Standard Normal

3

So Far, & The Future…Chapter 8

Inferential statistics: determine something about a population based only on the sample

Sampling A tool used to infer something about the

population Talk about 3 probability sampling methods

Construct a Distribution Of The Sample Mean Sample means tend to cluster around the

population mean Central Limit Theorem

Shape of the Distribution Of The Sample Mean tends to follow the normal probability distribution

4

A Sample Is The Only Feasible Way To Learn About A Population

1.The physical impossibility of checking all items in the population Example:

Can’t count all the fish in the ocean

2.The cost of studying all the items in a population Example:

General Mills hires firm to test a new cereal: Sample test: cost ≈ $40,000 Population test: cost ≈ $1,000,000,000

5

A Sample Is The Only Feasible Way To Learn About A Population

3. Contacting the whole population would often be time-consuming

Political polls can be completed in one or two days

Polling all the USA voters would take nearly 200 years!

4. The destructive nature of certain tests Examples:

Film from Kodak Seeds from Burpee

6

A Sample Is The Only Feasible Way To Learn About A Population

5. The sample results are usually adequate It is more than likely that the additional

accuracy of testing the whole population would not add a significant amount of improvement to the sample results

Example: Consumer price index constructed from a

sample is an excellent estimate for a consumer price index that could be constructed from the population

7

Probability Sampling

A sample selected in such a way that each item or person in the population has a known (nonzero) likelihood of being included in the sample

Known chance of being selected

8

Probability Sampling

Some of the methods used to select a sample:

1. Simple Random Sampling

2. Systematic Random Sampling

3. Stratified Random Sampling

• There is no “best” method of selecting a probability sample from a population of interest

• There are entire books devoted to sampling theory and design

9

Nonprobability Sample

In nonprobability sampling, inclusion in the sample is based on the judgment of the person selecting the sample

Nonprobability sampling can lead to biased results

10

Simple Random Sampling

A sample selected so that each item or person in the population has the same chance of being included

Example:1. Names of classmates in a hat, mix up names,

select until sample size, “n” is reached

2. Using a table of random numbers to select a sample from a population1. Appendix

11

Using A Table Of Random Variables To Prevent Bias In Selecting A Sample To

Represent A Population: Example:

Here at Highline, select 50 students at random to fill out questionnaire about tenured faculty performance

Steps:1. Use last four numbers of student ID2. Select random method to select starting point in random

number table Close eyes and point Month/day

3. Use first four numbers in table and match to last four in student ID If first four numbers in table do not match, move to next

This will give us a list of students that will constitute a sample with size 50

12

Select 50 Students At Random

If you encounter one that is in the table, but there is no corresponding student id, skip it

0.636222 0.878903 0.29908 0.054894 0.296277 0.077443 0.969701 0.361119 0.156304 0.8882440.518021 0.086118 0.492953 0.896894 0.269441 0.609022 0.554378 0.477563 0.307487 0.2743360.069223 0.051821 0.571346 0.399572 0.059926 0.825418 0.556095 0.895135 0.646497 0.4885570.686897 0.058912 0.893545 0.492101 0.420565 0.54019 0.37986 0.267845 0.74116 0.5263140.701841 0.441475 0.100799 0.659015 0.434216 0.785874 0.508761 0.337937 0.815675 0.7938050.370206 0.442265 0.883327 0.761954 0.884567 0.62314 0.554989 0.980878 0.227214 0.4509620.169599 0.943098 0.440034 0.495535 0.142257 0.139922 0.928234 0.249725 0.519689 0.7891460.87561 0.257012 0.655714 0.396071 0.426556 0.518622 0.679833 0.849866 0.200084 0.107449

0.330619 0.726719 0.140657 0.552255 0.99422 0.673014 0.972124 0.707176 0.934343 0.4933930.001197 0.773857 0.120766 0.054817 0.45625 0.545753 0.872873 0.611231 0.903448 0.6413840.412333 0.946429 0.668976 0.269024 0.959042 0.84274 0.256318 0.003595 0.043683 0.6878070.443448 0.195945 0.327642 0.864876 0.723678 0.990872 0.129111 0.122207 0.763613 0.464890.513373 0.821483 0.245097 0.627691 0.477342 0.961672 0.970896 0.831304 0.200614 0.9443490.593301 0.154681 0.904013 0.787415 0.10701 0.878112 0.990803 0.371747 0.717331 0.8570680.594944 0.56642 0.601252 0.408146 0.536719 0.015435 0.286428 0.113549 0.624948 0.8307020.259376 0.707408 0.363145 0.811295 0.686028 0.230584 0.179304 0.187189 0.136974 0.5864570.925766 0.928971 0.224099 0.862589 0.337782 0.677981 0.127947 0.202852 0.551529 0.8182260.973137 0.020498 0.965596 0.726818 0.305371 0.070409 0.639576 0.972905 0.644085 0.367109

Student Id #: 5180, 8611, 4929...

13

Systematic Random Sampling: The items or individuals of the population are arranged in

some order Invoice number Date Alphabetically Social security number

A random starting point is selected and then every kth member of the population is selected for the sample By starting randomly, all items have the same likelihood of being

selected for the sample Example: Audit Invoices for accuracy, start with 43rd

invoice and select every 20th invoice and check for accuracy

This method should not be used if there is a pattern to the population, or else you could get biased sample Example

14

Under Certain Conditions A Systematic Sample May Produce Biased Results

Inventory Count Problem:

Stacked bins with faster moving parts at the bottom

Start with 1st bin and count inventory for accuracy in every 3rd bin (may result in biased sample)

Simple random sampling would be better for this situation

6 7 18 19

5 8 17 20

4 9 16 21

3 10 15 22

2 11 14 23

1 12 13 24

Slow-moving bins = 4

moderately fast-moving bins = 16

Fast-moving bins = 2

Sample size = 8

Slow# Selected % of Sample (2/8) % Slow to Total (4/24)

2 25.00% 16.67%Slow

Mod

# Selected % of Sample (4/8) % Mod to Total (16/24)4 50.00% 66.67%M

od

Fast # Selected % of Sample (2/8) % Fast to Total (4/24)

2 25.00% 16.67%Fast

15

Stratified Random Sampling

A population is first divided into subgroups, called strata, and a sample is selected from each stratum

Advantage of stratified random sampling: Guarantees representation from each

subgroup

16

Proportional Sample Is Selected6 7 18 19

5 8 17 20

4 9 16 21

3 10 15 22

2 11 14 23

1 12 13 24

Slow-moving bins = 4

moderately fast-moving bins = 16

Fast-moving bins = 4

Sample Size, n = 8

# in Bin % Slow to total (4/24)4 16.67% 0.1667*8 = 1.3336

# in Bin % Mod to total (16/24)16 66.67% 0.6667*8 = 5.3336

# in Bin % Fast to total (4/24)4 16.67% 0.1667*8 = 1.3336

Bins selected from stratum (0.1667*8)

Bins selected from stratum (0.6667*8)

Bins selected from stratum (0.1667*8)

► 1

► 6

► 1

17

Cluster Sampling First:

A population is divided into primary units Second:

Primary units are selected at random (not all primary units will be selected)

Third: Samples are selected from the primary units

Employed to reduce the cost of sampling a population scattered over a large geographic area

Textbook shows geographic picture

18

Sampling Error

Will the mean of a sample always be equal to the population mean?

No! There will usually be some error: The difference between a sample statistic

and its corresponding population parameter Examples:

Xbar – μs – σ

19

Sampling Error

These sampling errors are due to chance The size of the error will vary from one

sample to the next So how can we make accurate predictions

based on samples??? Answer:

1. Sampling Distribution Of The Sample Meanand

2. The Central Limit Theorem

20

Sampling Distribution Of The Sample Mean

A probability distribution of all possible sample means of a given sample size Take a bunch of samples from the same

population Calculate the mean for each and plot all the

means

21

n = 36 Sample 1

Sample 1 Xbar 1

n = 36 Sample 2

Sample 2 Xbar 2

n = 36 Sample 3

Sample 3 Xbar 3

.

.

.

.

.

.n = 36 Sample n

Sample n Xbar n

Plot All Xbar

Sample 1

Sample 2

Sample 3

.

.

.

Sample n

n = 36 Sample 1

Sample 1 Xbar 1

n = 36 Sample 2

Sample 2 Xbar 2

n = 36 Sample 3

Sample 3 Xbar 3

.

.

.

.

.

.n = 36 Sample n

Sample n Xbar n

Construct Sampling Distribution Of The Sample Mean

22

Construct Sampling Distribution Of Sample Mean

1. Take many random samples of size “n” from a large population

2. Calculate the mean for each sample3. Plot all means on graph (frequency polygon)4. You would see that the curve looks normal! Textbook has good example

In particular: It shows how even if the population yields a skewed

probability distribution, the distribution of sample means will be approximately normal

Population mean = mean of the distribution of the sample mean

23

Plot Distribution Of The Sample Mean (Approximately Normal)

Mean of the Distribution of the Sample Means

SD of the Distribution of the Sample MeansX

X

-3 -2 -1 0 1 2 3 z

Sample Mean Xbar

Sampling Distribution Of The Sample Mean

In Class ConstructionOf

Distribution of Sample MeansAnd Prove that

µ = µbar

24

Central Limit Theorem

• If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples

• If population distribution is symmetrical but not normal, the distribution will converge toward normal when n > 10

• Skewed or thick-tailed distributions converge toward normal when n > 30

• Look at picture on page 265

25

Central Limit Theorem We can reason about the distribution of the

sample mean with absolutely no information about the shape of the original distribution from which the sample is taken The central limit theorem is true for all distributions

Central Limit Theorem will help us with: Chapter 9

Confidence intervals

Chapter 10 Tests of Hypothesis

26

Mean Of The Distribution Of The Sample Mean

If we are able to select all possible samples of a particular size from a given population, then the mean of the distribution of the sample mean will exactly equal the population mean:

Mean of the Distribution of the Sample MeansX

Mean of the Distribution of the Sample MeansX

Even if we do not select all possible samples, they will be approximately equal:

Sum of all sample means

Total number of samplesX

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Standard Deviation Of The Sampling Distribution Of The Sample Mean

(Standard Error Of The Mean)

There is less dispersion in the sampling distribution of the sample mean than in the population (each value is an average!!)

SD of the Sampling Distribution of the Sample MeanX n

σ = population standard deviation n = sample size When we increase “n” the standard deviation of

the sample will decrease

28

Central Limit Theorem

• Use the Central Limit Theorem to find probabilities of selecting possible sample means from a specified population

If the population is known to follow a normal distribution, or, n > 30…

We need our z-scores…

29

Z-Scores To determine the probability a sample

mean falls within a particular region, use:

n

Xz

Sampling error

Standard error of sampling distribution of the sample mean

We are interested in the distribution Xbar, the sample mean,instead of X

30

Business Decisions Example 1 History for a food manufacturer shows the

weight for a Chocolate Covered Sugar Bombs (popular breakfast cereal) is: μ = 14 oz. σ = .4 oz.

If the morning shift sample shows: Xbar = 14.14 oz. n = 30

Is this sampling error reasonable, or do we need to shut down the filling operations?

31

Business Decisions Example 1

14.14 14

.4 / 30

1.91703

Xz

n

μ = 14 oz.σ = 0.4 oz.

Xbar = 14.14 oz.

n = 30z = 1.917

Sugar Bombs Info.Table shows an area of .4726

.5 - .4726 = .0274

It is unlikely that we could sample and get this weight, so we must investigate the box filling equipment

In the distribution of sampling means, it is unlikely of getting a sample with 14.14 oz.

Suppose the mean selling price of a gallon of gasoline in the United States is $1.30. (μ) Further, assume the distribution is positively skewed, with a standard deviation of $0.28 (sigma). What is the probability of selecting a sample of 35 gasoline stations (n = 35)

and finding the sample mean within $.08?

$1.38 $1.301.69

$0.28 35

Xzsigma n

$1.22 $1.301.69

$0.28 35

Xzsigma n

Step One : Find the z-values corresponding to $1.22 and $1.38. These are the two points within $0.08 of the population mean.

9090.)4545(.2)69.169.1( zP

Step Two: determine the probability of a z-value between -1.69 and 1.69.

We would expect about 91 percent of the sample

means to be within $0.08 of the population mean.