1. Understanding Sampling Distributions: Statistics as Random Variables
Chapter 5_Part B
Understanding Sampling Distributions: Statistics as Random Variables
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2. EXPECTATIONS SAMPLING DISTRIBUTION FOR ONE SAMPLE PROPORTION;
SAMPLING DISTRIBUTION FOR ONE SAMPLE MEAN;
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3. Parameters, Statistics, and Statistical Inference
A statistic is a numerical value computed from a sample. Its value may differ for different samples. e.g. sample mean , sample standard deviation s, and sample proportion .
A parameter is a numerical value associated with a population. Considered fixed and unchanging. e.g. population mean m, population standard deviation s, and population proportion p.
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4. Inferential Statistics
Involves
Estimation
Hypothesis Testing
Purpose
Make decisions about population characteristics
Population?
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5. Statistical Inference
Statistical Inference: making conclusions about population parameters on basis of sample statistics.
Two most common procedures:
Confidence intervals: an interval of values that the researcher is fairly sure will cover the true, unknown value of the population parameter.
Hypothesis tests: uses sample data to attempt to reject a hypothesis about the population.
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6. Sampling Distribution For One Sample Proportion
PROBLEM FORMULATION: SUPPOSE THAT p IS AN UNKNOWN PROPORTION OF ELEMENTS OF A CERTAIN TYPE S IN A POPULATION.
EXAMPLES
PROPORTION OF LEFT - HANDED PEOPLE;
PROPORTION OF HIGH SCHOOL STUDENTS WHO ARE FAILING A READING TEST;
PROPORTION OF VOTERS WHO WILL VOTE FOR MR. X.
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7. Notation
Estimating the proportion falling into a category of a categorical variable.
Population parameter: p = proportion in the population falling into that category.
Sample estimate: = proportion in the sample falling into that category.
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8. Estimation of p
TO ESTIMATE p, WE SELECT A SIMPLE RANDOM SAMPLE (SRS), OF SIZE SAY, n = 1000, AND COMPUTE THE SAMPLE PROPORTION.
SUPPOSE THE NUMBER OF THE TYPE WE ARE INTERESTED IN, IN THIS SAMPLE OF n = 1000 IS x = 437. THEN THE SAMPLE PROPORTION
IS COMPUTED USING THE FORMULA
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9. Estimation of p
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10. WHAT IS THE ERROR OF ESTIMATION?
THAT IS, WHAT IS
IS THERE A MODEL (PROBABILITY DISTRIBUTION MODEL) THAT CAN HELP US FIND THE BEST ESTIMATE OF THE TRUE PROPORTION OF p?
LET’S START THE ANALYSIS BY FIRST ANSWERING THE SECOND QUESTION.
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11. THE APPROACH
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12. Sample Distribution Table
Sample of size n
X = Number of type S
Sample 1
Sample 2
Sample k
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13. Sampling Distributions
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14. REMARKS ON OBSERVING THE HISTOGRAM
THE HISTOGRAM ABOVE IS AN EXAMPLE OF WHAT WE WOULD GET IF WE COULD SEE ALL THE PROPORTIONS FROM ALL POSSIBLE SAMPLES. THAT DISTRIBUTION HAS A SPECIAL NAME. IT IS CALLED THE SAMPLING DISTRIBUTION OF THE PROPORTIONS.
OBSERVE THAT THE HISTOGRAM IS UNIMODAL, ROUGHLY SYMMETRIC, AND IT’S CENTERED AT p.
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15. Example – Gene For A Disease
Suppose (unknown to us) 40% of a population carry the gene for a disease, (p = 0.40).
We will take a random sample of 25 people from this population and count X = number with gene.
Although we expect (on average) to find 10 people (40%) with the gene, we know the number will vary for different samples of n = 25.
In this case, X is a binomial random variable with n = 25 and p = 0.4.
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16. Many Possible Samples Four possible random samples of 25 people: Note:
Sample 1: X =12, proportion with gene =12/25 = 0.48 or 48%.
Sample 2: X = 9, proportion with gene = 9/25 = 0.36 or 36%.
Sample 3: X = 10, proportion with gene = 10/25 = 0.40 or 40%.
Sample 4: X = 7, proportion with gene = 7/25 = 0.28 or 28%.
Note:
Each sample gave a different answer, which did not always match the population value of p.
Although we cannot determine whether one sample will accurately reflect the population, statisticians have determined what to expect for most possible samples.
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17. Example – Gene For A Disease
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18. Sampling Distribution For One Sample Proportion
Statistics as Random Variables
Each new sample taken 
value of the sample statistic will change.
The distribution of possible values of a statistic for repeated samples of the same size from a population is called the sampling distribution of the statistic.
Many statistics of interest have sampling distributions that are approximately normal distributions
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19. WHAT THEN IS THE APPROPRIATE PROBABILITY MODEL?
ANSWER: IT IS AMAZING AND FORTUNATE THAT A NORMAL MODEL IS JUST THE RIGHT ONE FOR THE HISTOGRAMS OF SAMPLE PROPORTIONS.
HOW GOOD IS THE NORMAL MODEL?
IT IS GOOD IF THE FOLLOWING ASSUMPTIONS AND CONDITIONS HOLD.
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20. ASSUMPTIONS AND CONDITIONS
INDEPENDENCE ASSUMPTION: THE SAMPLED VALUES MUST BE INDEPENDENT OF EACH OTHER.
SAMPLE SIZE ASSUMPTION: THE SAMPLE SIZE, n, MUST BE LARGE ENOUGH
REMARK: ASSUMPTIONS ARE HARD – OFTEN IMPOSSIBLE TO CHECK. THAT’S WHY WE ASSUME THEM. GLADLY, SOME CONDITIONS MAY PROVIDE INFORMATION ABOUT THE ASSUMPTIONS.
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21. CONDITIONS
RANDOMIZATION CONDITION: THE DATA VALUES MUST BE SAMPLED RANDOMLY. IF POSSIBLE, USE SIMPLE RANDOM SAMPLING DESIGN TO SAMPLE THE POPULATION OF INTEREST.
10% CONDITION: THE SAMPLE SIZE, n, MUST BE NO LARGER THAN 10% OF THE POPULATION OF INTEREST.
SUCCESS/FAILURE CONDITION: THE SAMPLE SIZE HAS TO BE BIG ENOUGH SO THAT WE EXPECT AT LEAST 10 SUCCESSES AND AT LEAST 10 FAILLURES. THAT IS,
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22. Sampling Distribution for a Sample Proportion (The Central Limit Theorem)
Let p = population proportion of interest or binomial probability of success.
Let = sample proportion or proportion of successes.
If numerous random samples or repetitions of the same size n are taken, the distribution of possible values of is approximately a normal curve distribution with
Mean = p
Standard deviation = s.d.( ) =
This approximate distribution is sampling distribution of .
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23. Estimating the Population Proportion from a Single Sample Proportion
In practice, we don’t know the true population proportion p, so we cannot compute the standard deviation of ,
s.d.( ) =
In practice, we only take one random sample, so we only have one sample proportion Replacing p with in the standard deviation expression gives us an estimate that is called the standard error of .
s.e.( ) = .
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24. More Examples for which Rule Applies
Election Polls: to estimate proportion who favor a candidate; units = all voters.
Television Ratings: to estimate proportion of households watching TV program; units = all households with TV.
Consumer Preferences: to estimate proportion of consumers who prefer new recipe compared with old; units = all consumers.
Testing ESP: to estimate probability a person can successfully guess which of 5 symbols on a hidden card; repeatable situation = a guess.
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25. EXAMPLE FROM PRACTICE SHEET
ASSUME THAT 30% OF STUDENTS AT A UNIVERSITY WEAR CONTACT LENSES
(A) WE RANDOMLY PICK 100 STUDENTS. LET REPRESENT THE PROPORTION OF STUDENTS IN THIS SAMPLE WHO WEAR CONTACTS. WHAT’S THE APPROPRIATE MODEL FOR THE DISTRIBUTION OF ? SPECIFY THE NAME OF THE DISTRIBUTION, THE MEAN, AND THE STANDARD DEVIATION. BE SURE TO VERIFY THAT THE CONDITIONS ARE MET.
(B) WHAT’S THE APPROXIMATE PROBABILITY THAT MORE THAN ONE THIRD OF THIS SAMPLE WEAR CONTACTS?
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26. SOLUTION
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27. EXAMPLE FROM PRACTICE SHEET
INFORMATION ON A PACKET OF SEEDS CLAIMS THAT THE GERMINATION RATE IS 92%. WHAT’S THE PROBABILITY THAT MORE THAN 95% OF THE 160 SEEDS IN THE PACKET WILL GERMINATE? BE SURE TO DISCUSS YOUR ASSUMPTIONS AND CHECK THE CONDITIONS THAT SUPPORT YOUR MODEL.
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28. SOLUTION
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29. Sampling Distribution for One Sample Mean
Suppose we want to estimate the mean weight loss for all who attend clinic for 10 weeks. Suppose (unknown to us) the distribution of weight loss is approximately N(8 pounds, 5 pounds).
We will take a random sample of 25 people from this population and record for each X = weight loss.
We know the value of the sample mean will vary for different samples of n = 25.
What do we expect those means to be?
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30. Familiar Examples Estimating the mean of a quantitative variable.
Example research questions:
What is the mean time that college students watch TV per day? What is the mean pulse rate of women?
Population parameter: m = population mean for the variable
Sample estimate: = sample mean for the variable
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31. Many Possible Samples Four possible random samples of 25 people: Note:
Sample 1: Mean = 8.32 pounds, standard deviation = 4.74 pounds.
Sample 2: Mean = 6.76 pounds, standard deviation = 4.73 pounds.
Sample 3: Mean = 8.48 pounds, standard deviation = 5.27 pounds.
Sample 4: Mean = 7.16 pounds, standard deviation = 5.93 pounds.
Note:
Each sample gave a different answer, which did not always match the population mean of 8 pounds.
Although we cannot determine whether one sample mean will accurately reflect the population mean, statisticians have determined what to expect for most possible sample means.
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32. The Normal Curve Approximation Rule for Sample Means (The Central Limit Theorem)
Let m = mean for population of interest. Let s = standard deviation for population of interest.
Let = sample mean.
If numerous random samples of the same size n are taken, the distribution of possible values of is approximately a normal curve distribution with
Mean = m
Standard deviation = s.d.( ) =
This approximate distribution is sampling distribution of .
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33. Standard Error of the Mean
In practice, the population standard deviation s is rarely known, so we cannot compute the standard deviation of ,
s.d.( ) =
In practice, we only take one random sample, so we only have the sample mean and the sample standard deviation s. Replacing s with s in the standard deviation expression gives us an estimate that is called the standard error of .
s.e.( ) =
For a sample of n = 25 weight losses, the standard deviation is s = 4.74 pounds. So the standard error of the mean is pounds.
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34. ASSUMPTIONS AND CONDITIONS
INDEPENDENCE ASSUMPTION: THE SAMPLED VALUES MUST BE INDEPENDENT OF EACH OTHER
SAMPLE SIZE ASSUMPTION: THE SAMPLE SIZE MUST BE SUFFICIENTLY LARGE.
REMARK: WE CANNOT CHECK THESE DIRECTLY, BUT WE CAN THINK ABOUT WHETHER THE INDEPENDENCE ASSUMPTION IS PLAUSIBLE.
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35. CONDITIONS
RANDOMIZATION CONDITION: THE DATA VALUES MUST BE SAMPLED RANDOMLY, OR THE CONCEPT OF A SAMPLING DISTRIBUTION MAKES NO SENSE. IF POSSIBLE, USE SIMPLE RANDOM SAMPLING DESIGN TO ABTAIN THE SAMPLE.
10% CONDITION: WHEN THE SAMPLE IS DRAWN WITHOUT REPLACEMENT (AS IS USUALLY THE CASE), THE SAMPLE SIZE, n, SHOULD BE NO MORE THAN 10% OF THE POPULATION.
LARGE ENOUGH SAMPLE CONDITION: IF THE POPULATION IS UNIMODAL AND SYMMETRIC, EVEN A FAIRLY SMALL SAMPLE IS OKAY. IF THE POPULATION IS STRONGLY SKEWED, IT CAN TAKE A PRETTY LARGE SAMPLE TO ALLOW USE OF A NORMAL MODEL TO DESCRIBE THE DISTRIBUTION OF SAMPLE MEANS
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36. Examples for which Rule Applies
Average Weight Loss: to estimate average weight loss; weight assumed bell-shaped; population = all current and potential clients.
Average Age At Death: to estimate average age at which left-handed adults (over 50) die; ages at death not bell-shaped so need n  30; population = all left-handed people who live to be at least 50.
Average Student Income: to estimate mean monthly income of students at university who work; incomes not bell-shaped and outliers likely, so need large random sample of students; population = all students at university who work.
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37. EXAMPLES FROM PRACTICE SHEET
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38. Increasing the Size of the Sample
Suppose we take n = 100 people instead of just 25. The standard deviation of the mean would be
s.d.( ) = pounds.
For samples of n = 25, sample means are likely to range between 8 ± 3 pounds => 5 to 11 pounds.
For samples of n = 100, sample means are likely to range only between 8 ± 1.5 pounds => 6.5 to 9.5 pounds.
Larger samples tend to result in more accurate estimates of population values than smaller samples.
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39. The Central Limit Theorem (CLT)
The Central Limit Theorem states that if n is sufficiently large, the sample means of random samples from a population with mean m and finite standard deviation s are approximately normally distributed with mean m and standard deviation
Technical Note: The mean and standard deviation given in the CLT hold for any sample size; it is only the “approximately normal” shape that requires n to be sufficiently large.
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40. Sampling Distribution for Any Statistic
Every statistic has a sampling distribution, but the appropriate distribution may not always be normal, or even approximately bell-shaped.
Construct an approximate sampling distribution for a statistic by actually taking repeated samples of the same size from a population and constructing a relative frequency histogram for the values of the statistic over the many samples.
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