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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
27
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.