1. Ch 8 ==> Statistics Is Fun!
Chapter 8 Goals
Explain why a sample is the only feasible way to learn about a population
Describe methods to select a sample:
Simple Random Sampling
Systematic Random Sampling
Stratified Random Sampling
Sampling Error
Sampling Distribution Of The Sample Mean
Central Limit Theorem
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2. Ch 8 ==> Statistics Is Fun!
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
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3. Ch 8 ==> Statistics Is Fun!
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
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4. A Sample Is The Only Feasible Way To Learn About A Population
Ch 8 ==> Statistics Is Fun!
A Sample Is The Only Feasible Way To Learn About A Population
The physical impossibility of checking all items in the population
Example:
Can’t count all the fish in the ocean
The cost of studying all the items in a population
General Mills hires firm to test a new cereal:
Sample test: cost ≈ $40,000
Population test: cost ≈ $1,000,000,000
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5. A Sample Is The Only Feasible Way To Learn About A Population
Ch 8 ==> Statistics Is Fun!
A Sample Is The Only Feasible Way To Learn About A Population
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!
The destructive nature of certain tests
Examples:
Film from Kodak
Seeds from Burpee
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6. A Sample Is The Only Feasible Way To Learn About A Population
Ch 8 ==> Statistics Is Fun!
A Sample Is The Only Feasible Way To Learn About A Population
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
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7. Ch 8 ==> Statistics Is Fun!
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
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8. Ch 8 ==> Statistics Is Fun!
Probability Sampling
Some of the methods used to select a sample:
Simple Random Sampling
Systematic Random Sampling
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
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9. Nonprobability Sample
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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
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10. Simple Random Sampling
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Simple Random Sampling
A sample selected so that each item or person in the population has the same chance of being included
Example:
Names of classmates in a hat, mix up names, select until sample size, “n” is reached
Using a table of random numbers to select a sample from a population
Appendix
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11. Ch 8 ==> Statistics Is Fun!
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:
Use last four numbers of student ID
Select random method to select starting point in random number table
Close eyes and point
Month/day
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
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12. Select 50 Students At Random
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Select 50 Students At Random
If you encounter one that is in the table, but there is no corresponding student id, skip it
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13. Systematic Random Sampling:
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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
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14. Ch 8 ==> Statistics Is Fun!
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
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15. Stratified Random Sampling
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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
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16. Proportional Sample Is Selected
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Proportional Sample Is Selected
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17. Ch 8 ==> Statistics Is Fun!
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
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18. Ch 8 ==> Statistics Is Fun!
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 – μ
s – σ
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19. Ch 8 ==> Statistics Is Fun!
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:
Sampling Distribution Of The Sample Mean and
The Central Limit Theorem
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20. Sampling Distribution Of The Sample Mean
Ch 8 ==> Statistics Is Fun!
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
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21. Construct Sampling Distribution Of The Sample Mean
Ch 8 ==> Statistics Is Fun!
Construct Sampling Distribution Of The Sample Mean
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22. Construct Sampling Distribution Of Sample Mean
Ch 8 ==> Statistics Is Fun!
Construct Sampling Distribution Of Sample Mean
Take many random samples of size “n” from a large population
Calculate the mean for each sample
Plot all means on graph (frequency polygon)
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
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23. Plot Distribution Of The Sample Mean (Approximately Normal)
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Plot Distribution Of The Sample Mean (Approximately Normal)
In Class Construction Of
Distribution of Sample Means
And Prove that
µ = µbar
Sampling Distribution Of The Sample Mean
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24. Ch 8 ==> Statistics Is Fun!
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
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25. Ch 8 ==> Statistics Is Fun!
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
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26. Mean Of The Distribution Of The Sample Mean
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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:
Even if we do not select all possible samples, they will be approximately equal:
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27. Ch 8 ==> Statistics Is Fun!
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!!)
σ = population standard deviation
n = sample size
When we increase “n” the standard deviation of the sample will decrease
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28. Ch 8 ==> Statistics Is Fun!
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…
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29. Ch 8 ==> Statistics Is Fun!
Z-Scores
To determine the probability a sample mean falls within a particular region, use:
Sampling error
Standard error of sampling distribution of the sample mean
We are interested in the distribution Xbar, the sample mean,
instead of X
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30. Business Decisions Example 1
Ch 8 ==> Statistics Is Fun!
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 = oz.
n = 30
Is this sampling error reasonable, or do we need to shut down the filling operations?
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31. Business Decisions Example 1
Ch 8 ==> Statistics Is Fun!
Business Decisions Example 1
Table shows an area of .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 oz.
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32. Ch 8 ==> Statistics Is Fun!
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?
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33. Ch 8 ==> Statistics Is Fun!
Step One : Find the z-values corresponding to $1.22 and $ These are the two points within $0.08 of the population mean.
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34. Ch 8 ==> Statistics Is Fun!
Step Two: determine the probability of a z-value between and 1.69.
We would expect about 91 percent of the sample means to be within $0.08 of the population mean.
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