Statistics- Random Sampling

8/3/2019 Statistics- Random Sampling

1/23

Slide

8-1

2/10/2012

Chapter 8

Random Sampling: Planning

Ahead for Data Gathering


2/23

Slide

8-2

2/10/2012

Random Sampling

The basis for statistical inference about a

population based on a sample

Example: Build restaurant in neighborhood?

Population: the collection of items you want tounderstand

Nitems. Example: all people in neighborhood

Sample: a smaller collection of population units

n items. Example: 100 neighborhood residents whoagree to be interviewed

Which 100 residents?

How to select?


3/23

Slide

8-3

2/10/2012

Terminology of Sampling

Representative Sample

Same percentages in sample as in population

Example: sample is representative if same percent:

work, are young/old, are single/married, etc

Biased Sample

Not representative of population in an important way

Example: sample is biased if too many retired people

Frame: Access to population (by number from 1 to N)

Example: from phone book:

1. Fred Jones

2. Mary Kipling

N. Carol Lewis


4/23

Slide

8-4

2/10/2012

Visualizing the Sampling Process

Here is a fairly (but not perfectly) representative sample

Note that neither open triangle was selected

Sample

Population


5/23

Slide

8-5

2/10/2012

Sampling Terminology (continued)

Sampling Without Replacement

No unit may appear more than once in the sample

After a unit is selected, it is removed from the population

before another population unit is drawn

Sampling With Replacement A unit can be represented more than once in the sample

After a unit is selected, it is replaced back in the population

before another unit is drawn

Census A sample that consists of the entire population

Often too expensive

Even when affordable, may not be cost-efficient


6/23

Slide

8-6

2/10/2012

Statistic and Parameter

Sample Statistic

Any number computed from sample data

A random variable. Known

Example: Average weekly food expenditures for 100 sampled

residents Random? Yes! Due to randomness of sample selection

Population Parameter

Any number computed for the entire population

A fixed number. Unknown

Example: mean weekly food expenditures for all 77,386

residents

Do we ever know this? NO!

But we estimate it (with error)


7/23

Slide

8-7

2/10/2012

Estimator and Estimate

Estimator

A sample statistic used to guess a population parameter

Example: Sample average for100 selected residents is an

estimator of the population mean of all 77,386 residents

Estimate [WRONG! Estimators are usually wrong. Often useful anyway]

The actual number computed from the data

Example: $33.91 is an estimate of neighborhood weekly food

expenditures per person

Estimation error Estimator minus population parameter. Unknown

Example: 33.91 35.69 = 1.78

Average, 100 residents Mean, all 77,386

residents (unknown)

Estimation error (unknown)


8/23

Slide

8-8

2/10/2012

Sampling Terminology (continued)

Unbiased Estimator

Correct on average. Neither systematically too high nor

too low

Hits the target on average although it may miss each time

Pilot Study

Small-scale test before full study is run

Test your questionnaire on a few people, change as needed,

and then use it on hundreds more

Sampling Distribution The probability distribution of a statistic computed

using data from a random sample


9/23

Slide

8-9

2/10/2012

Random Sample

Provides a foundation for statistical inference

A random sample must satisfy:

1. Each population unit must have an equal chance of

being selected

This helps assure representation, because all units in the

population are equally accessible

2. Units must be selected independently of one another

This guarantees that each item to be selected will bring new,

independent information

Properties

Sample is representative of population (on average)

Statistical inference will use randomness of the sample


10/23

Slide

8-10

2/10/2012

Random Samples of5 from 36

On 6by 6 grid

5 squares shaded in, selected at random, one at a time

Without replacement

Note that adjacent (touching) squares can be selected

To exclude them would make selection non-independent

Perhaps you see systematic patterns

They are not there by design, but by coincidence

But a checkerboard pattern would very likely notbe random


11/23

Slide

8-11

2/10/2012

Selecting a Random Sample

Use Table of Random Digits

Establish the frame (population units from 1 to N)

Decide starting place in table of random digits

Read random digits in groups

e.g., ifN= 5,281, then use groups of4 digits (Nhas 4 digits)

Include number group

if it is from 1 to N, and has not yet been chosen

Shuffle the Population (Spreadsheet)

Arrange the population items in a column from 1 to N

Put random numbers in an adjacent column: =RAND()

Sort population items in order by random numbers


12/23

Slide

8-12

2/10/2012

Table ofRandom Digits

For example

Starting in row 21, column 3

We find 52794, then 01466

1 2 3 4 5 6 7 8 9 10

1 51449 39284 85527 67168 91284 19954 91166 70918 85957 194922 16144 56830 67507 97275 25982 69294 32841 20861 83114 12531

3 48145 48280 99481 13050 81818 25282 66466 24461 97021 210724 83780 48351 85422 42978 26088 17869 94245 26622 48318 73850

5 95329 38482 93510 39170 63683 40587 80451 43058 81923 970726 11179 69004 34273 36062 26234 58601 47159 82248 95968 99722

7 94631 52413 31524 02316 27611 15888 13525 43809 40014 306678 64275 10294 35027 25604 65695 36014 17988 02734 31732 29911

9 72125 19232 10782 30615 42005 90419 32447 53688 36125 2845610 16463 42028 27927 48403 88963 79615 41218 43290 53618 68082

11 10036 66273 69506 19610 01479 92338 55140 81097 73071 61544

12 85356 51400 88502 98267 73943 25828 38219 13268 09016 77465

13 84076 82087 55053 75370 71030 92275 55497 97123 40919 57479

14 76731 39755 78537 51937 11680 78820 50082 56068 36908 55399

15 19032 73472 79399 05549 14772 32746 38841 45524 13535 03113

16 72791 59040 61529 74437 74482 76619 05232 28616 98690 24011

17 11553 00135 28306 65571 34465 47423 39198 54456 95283 54637

18 71405 70352 46763 64002 62461 41982 15933 46942 36941 93412

19 17594 10116 55483 96219 85493 96955 89180 59690 82170 7764320 09584 23476 09243 65568 89128 36747 63692 09986 47687 46448

21 81677 62634 52794 01466 85938 14565 79993 44956 82254 6522322 45849 01177 13773 43523 69825 03222 58458 77463 58521 07273

23 97252 92257 90419 01241 52516 66293 14536 23870 78402 4175924 26232 77422 76289 57587 42831 87047 20092 92676 12017 43554

25 87799 33602 01931 66913 63008 03745 93939 07178 70003 1815826 46120 62298 69126 07862 76731 58527 39342 42749 57050 91725

27 53292 55652 11834 47581 25682 64085 26587 92289 41853 3835428 81606 56009 06021 98392 40450 87721 50917 16978 39472 23505

29 67819 47314 96988 89931 49395 37071 72658 53947 11996 6463130 50458 20350 87362 83996 86422 58694 71813 97695 28804 58523

31 59772 27000 97805 25042 09916 77569 71347 62667 09330 02152

32 94752 91056 08939 93410 59204 04644 44336 55570 21106 76588

33 01885 82054 45944 55398 55487 56455 56940 68787 36591 29914

34 85190 91941 86714 76593 77199 39724 99548 13827 84961 76740

35 97747 67607 14549 08215 95408 46381 12449 03672 40325 77312

36 43318 84469 26047 86003 34786 38931 34846 28711 42833 93019

37 47874 71365 76603 57440 49514 17335 71969 58055 99136 7358938 24259 48079 71198 95859 94212 55402 93392 31965 94622 11673

39 31947 64805 34133 03245 24546 48934 41730 47831 26531 0220340 37911 93224 87153 54541 57529 38299 65659 00202 07054 40168

41 82714 15799 93126 74180 94171 97117 31431 00323 62793 1199542 82927 37884 74411 45887 36713 52339 68421 35968 67714 05883

43 65934 21782 35804 36676 35404 69987 52268 19894 81977 8776444 56953 04356 68903 21369 35901 86797 83901 68681 02397 55359

45 16278 17165 67843 49349 90163 97337 35003 34915 91485 3381446 96339 95028 48468 12279 81039 56531 10759 19579 00015 22829

47 84110 49661 13988 75909 35580 18426 29038 79111 56049 9645148 49017 60748 03412 09880 94091 90052 43596 21424 16584 67970

49 43560 05552 54344 69418 01327 07771 25364 77373 34841 7592750 25206 15177 63049 12464 16149 18759 96184 15968 89446 07168

Table 8.2.1

19 17594 10116 55483 96219 85493 96955

20 09584 23476 09243 65568 89128 36747

21 81677 62634 52794 01466 85938 14565

22 45849 01177 13773 43523 69825 03222

23 97252 92257 90419 01241 52516 66293

24 26232 77422 76289 57587 42831 87047


13/23

Slide

8-13

2/10/2012

Example: Sample Selection

Select random sample using random number table

Of size n = 4 from a population of size N= 861, starting

at row 21, column 3 of the Table of Random Digits

Start with random digits

52794 01466 85938 14565 79993

Group by 3 (because N= 861 has 3 digits)

527 940 146 685 938 145 657

Omit 000, and also 862, 863, , 999

Omit duplicates, until n = 4 are obtained in the sample

527 146 685 145

The random sample includes the following population units

(numbered by frame):

527, 146, 685, 145


14/23

Slide

8-14

2/10/2012

Visualizing the Sampling Process

The process:

Sample n, compute statistic

is the same as the process:

Choose 1 from sampling distribution

Sample

n units

Statistic

(estimator)

Sample

n units

Statistic

(estimator)

Sample

n units

Statistic

(estimator)

Sample

n units

Statistic

(estimator)

The Population

A histogram of these imagined

values represents the sampling

distribution of this statistic


15/23

Slide

8-15

2/10/2012

Central Limit Theorem

Helps you find probabilities for an average ofn

independent individuals by giving you

The mean

The standard deviation

The right to use the normal probability tables:

Ifn is large, then the average is approximately normal,

even if individuals are skewed

Works for totals also by giving you

The mean

The standard deviation

The right to use the normal probability tables fortotal

Q!Q!QXX

nX

/W!W

X

Q!Q ntotal

ntotal

W!W

X


16/23

Slide

8-16

2/10/2012

Central Limit Theorem (continued)

Individuals in population

Highly non-normal distribution

Mean Q, standard deviation W

Averages ofn = 3 individuals

Non-normal, but less so

Same mean Q

Lower std. deviation:

Averages ofn = 10 individuals

Close to normal

Same mean Q

Lower std. deviation

0

300

0 5 10

0

100

200

0 5 10

0

100

0 5 10

3/W!WX

10/W!WX

Q

Q

Q


17/23

Slide

8-17

2/10/2012

What good to us is ?

Why bother choosing a larger sample to estimate Q?

Even sampling just n= 1 (or a few) we have an unbiased

estimator that is, on average, equal to Q

But the error can be very large! When we sample n= 100 orn= 1,000, what we gain for our

trouble is the LOWER ERROR , which says that

our estimator, , is closer to the unknown population mean Q

It tells us how the variability of the sample average

is related to the variability of individuals in thepopulation

This will be useful soon in defining thestandard error

Central Limit Theorem (continued)

nX

/W!W

nX

/W!W

X

X


18/23


19/23

Slide

8-19

2/10/2012

Example (continued) The problem: Find the probability that the totalweight of a

box exceeds 215 ounces

By central limit theorem, total weight is

approximately normal with

Standardize, using CLT:

Find

Draw picture, use normal table

Answer is 0.32

210730 !v!Q!Q ntotal 95.10302 !!W!W ntotal

"!

"

W

Q46.0

normal

standardProb

95.10

210215Prob

total

totaltotal

-3 -2 -1 0 1 2 3

0.46


20/23

Slide

8-20

2/10/2012

Standard Error

Definition: the Estimated Standard Deviation (ofthe sampling distribution) of a statistic

Standard Error of the Average

Indicates approximately how far the sample average is

from the population mean Q

Advantage: can be computed using sample data

Standard Deviation of the Average

Also indicates approximately how far the sample average

is from the population mean Q

Problem: cannot be computed without population parameters

X

n

SS

X!

n

X

W

!W

X


21/23

Slide

8-21

2/10/2012

Standard Error for Binomial

Standard Error ofp (binomial percent) is

Indicates approximately how far the sample estimatep is fromits population value T

Advantage: can be computed using sample data

Standard Deviation ofp is

Also indicates approximately how far the sample estimatep is

from its population value T

Problem: cannot be computed without population parameters

n

ppSp

)1( !

np

)1( TT!W


22/23

Slide

8-22

2/10/2012

Example of Standard Error

n= 100 Sample Size: Number of people in sample

= $23.91 Sample Average: Typical expenditure for people

in sample. An estimate of typical expenditure Q

for people in the population

S= $11.49 Standard Deviation: Individuals in the sampleare approximately S= $11.49 away from =

$23.91. Individuals in the population are

approximately W=? away from Q=?

Standard Error: Thesample average is approximately

from Q=?

X

X

15.1$100/49.11/ !!! nSSX

15.1$!X

S


23/23

Slide

8-23

2/10/2012

Example: Binomial Standard Error

X= 83 out ofn = 268 interviewed said they would buy theproduct

p = X/n = 83/268 = 0.3097 or31.0% is the sample percent, an

estimate ofT, the (unknown) population percent

How far is the (known)p = 31.0% from the (unknown) T =?Answer: aboutone standard error

The observed sample percentage,

31.0%, is approximately 2.82%

(in percentage points) away from

the unknown population

percentage T

%82.20282.0268

)3097.01(3097.0

)1(

!!

v!

!

n

ppSp

Statistics- Random Sampling

Documents

Transcript of Statistics- Random Sampling