1. Chapter 7 Sampling Distributions
Basic Business Statistics 10th Edition
Chapter 7
Sampling Distributions
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc..

2. Learning Objectives In this chapter, you learn:
The concept of the sampling distribution
To compute probabilities related to the sample mean and the sample proportion
The importance of the Central Limit Theorem
To distinguish between different survey sampling methods
To evaluate survey worthiness and survey errors
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

3. Chapter 7 On-Line Problem Assignment (using MathXL)
From Textbook Chapter 7
Problem 7.17
Problem 7.25
Problem 7.21
Problem 7.29
Problem 7.33
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4. Login Web Address for MathXL/MyStatLab On-Line Problems
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5. Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
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6. Sampling Distributions
A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population
It is the probability distribution of a statistic
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7. If the Population is Normal
If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with
and
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8. Sampling Distributions
A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population
It is the probability distribution of a statistic
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9. Sampling Distribution of the Mean
Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
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10. Sampling Distribution of the Mean
When the population is normally distributed
Shape: Regardless of sample size, the distribution of sample means will be normally distributed.
Center: The mean of the distribution of sample means is the mean of the population. Sample size does not affect the center of the distribution.
Spread: The standard deviation of the distribution of sample means, or the standard error, is . n x s =
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11. Z-value for Sampling Distribution of the Mean
Z-value for the sampling distribution of :
where: = sample mean
= population mean
= population standard deviation
n = sample size
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12. Sampling Distribution Properties
Normal Population Distribution
(i.e is unbiased )
Normal Sampling Distribution
(has the same mean)
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13. Sampling Distribution Properties
(continued)
As n increases,
decreases
Larger sample size
Smaller sample size
(i.e is consistent )
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14. If the Population is not Normal
We can apply the Central Limit Theorem:
Even if the population is not normal,
…sample means from the population will be approximately normal as long as the sample size is large enough.
Properties of the sampling distribution:
and
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15. Central Limit Theorem
the sampling distribution becomes almost normal regardless of shape of population
As the sample size gets large enough… n↑
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16. Central Limit Theorem
According to the Central Limit Theorem (CLT), the larger the sample size (n 30), the more normal the distribution of sample means becomes. The CLT is central to the concept of statistical inference because it permits us to draw conclusions about the population based strictly on sample data without having knowledge about the distribution of the underlying population.
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17. Sampling Distribution of the Mean
When the population is not normally distributed
Shape: When the sample size taken from such a population is sufficiently large, the distribution of its sample means will be approximately normally distributed regardless of the shape of the underlying population those samples are taken from. According to the Central Limit Theorem, the larger the sample size, the more normal the distribution of sample means becomes.
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18. If the Population is not Normal
(continued)
Population Distribution
Sampling distribution properties:
Central Tendency
Sampling Distribution
(becomes normal as n increases)
Variation
Larger sample size
Smaller sample size
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19. How Large is Large Enough?
For most distributions, n  30 will give a sampling distribution that is nearly normal
For fairly symmetric distributions, n > 15
For normal population distributions, the sampling distribution of the mean is always normally distributed
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20. Example
Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.
What is the probability that the sample mean is between 7.8 and 8.2?
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21. Example
(continued)
Solution:
Even if the population is not normally distributed, the central limit theorem can be used (n > 30)
… so the sampling distribution of is approximately normal
… with mean = 8
…and standard deviation
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22. Example Solution (continued): (continued) Z X Population Distribution
Sampling Distribution
Standard Normal Distribution ? ? ? ? ? ? ? ? ? ?
Sample
Standardize ? ? Z X
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23. Table E.14 on page 835 of Text
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24. Sample Problem
You’re an operations analyst for Sprint-Nextel. Long-distance telephone calls are normally distributed with  = 8 min. &  = 2 min. If you select random samples of 25 calls, what percentage of the sample means would be between 7.8 & 8.2 minutes?
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25. Standardized Normal Distribution
Example:
Standardized Normal Distribution
Sampling Distribution
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26. Another Example
Example: When a production machine is properly calibrated, it requires an average of 25 seconds per unit produced, with a standard deviation of 3 seconds. For a simple random sample of n = 36 units, the sample mean is found to be 26.2 seconds per unit. When the machine is properly calibrated, what is the probability that the mean for a simple random sample of this size will be at least 26.2 seconds?
Standardized sample mean:
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27. Sampling Distribution of the Proportion
Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
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28. Population Proportion
Categorical Variable
E.g., Gender, Voted for Bush, College Degree
Proportion of Population Having a Certain Characteristic (p)
Sample Proportion Provides an Estimate of p
If Two Outcomes, X Has a Binomial Distribution
Possess or do not possess characteristic
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29. Population Proportions
π = the proportion of the population having
some characteristic
Sample proportion ( p ) provides an estimate
of π:
0 ≤ p ≤ 1
p has a binomial distribution
(assuming sampling with replacement from a finite population or without replacement from an infinite population)
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30. Sampling Distribution of p
Approximated by a normal distribution if:
where
and
Sampling Distribution
P( ps) .3 .2 .1 p
(where π = population proportion)
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31. Sampling Distribution of the Proportion
When the sample proportion of successes in a sample of n trials is p,
Center: The center of the distribution of sample proportions is the center of the population, p.
Spread: The standard deviation of the distribution of sample proportions, or the standard error, is
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32. Z-Value for Proportions
Standardize p to a Z value with the formula:
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33. Example
If the true proportion of voters who support Proposition A is π = 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45?
i.e.: if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45) ?
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34. Example if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ? Find :
(continued)
if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45) ?
Find :
Convert to standard normal:
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35. Standardized Normal Distribution
Example
(continued)
if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45) ?
Use standard normal table: P(0 ≤ Z ≤ 1.44) =
Standardized Normal Distribution
Sampling Distribution
0.4251
Standardize
0.40
0.45
1.44 p Z
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36. Another Proportion Example
Example: The campaign manager for a political candidate claims that 55% of registered voters favor the candidate over her strongest opponent. Assuming that this claim is true, what is the probability that in a simple random sample of 300 voters, at least 60% would favor the candidate over her strongest opponent?
p = 0.55, p = 0.60, n = 300
Standardized sample proportion:
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37. Sample Problem
You’re manager of a bank. 40% of depositors have multiple accounts. You select a random sample of 200 customers. What is the probability that the sample proportion of depositors with multiple accounts would be between 40% & 43% ?
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38.  Sampling Distribution of Proportion Solution P(.40  p  .43) Z = = =
0.87
Sampling Distribution
Standardized -
Normal Distribution
p = .0346
 = 1
.3078 p Z
p =
 =
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39. Standardized Normal Distribution
Example:
Standardized Normal Distribution
Sampling Distribution
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40. Sampling Distribution of the Proportion
When the sample proportion of successes in a sample of n trials is p,
Center: The center of the distribution of sample proportions is the center of the population, p.
Spread: The standard deviation of the distribution of sample proportions, or the standard error, is
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41. Reasons for Drawing a Sample
Less time consuming than a census
Less costly to administer than a census
Less cumbersome and more practical to administer than a census of the targeted population
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42. Types of Samples Used Nonprobability Sample Probability Sample
Items included are chosen without regard to their probability of occurrence
Probability Sample
Items in the sample are chosen on the basis of known probabilities
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43. Non-Probability Samples
Types of Samples Used
(continued)
Samples
Non-Probability Samples
Probability Samples
Simple
Random
Stratified
Judgement
Chunk
Cluster
Systematic
Quota
Convenience
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44. Probability Sampling
Items in the sample are chosen based on known probabilities
Probability Samples
Simple
Random
Systematic
Stratified
Cluster
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45. Simple Random Samples
Every individual or item from the frame has an equal chance of being selected
Selection may be with replacement or without replacement
Samples obtained from table of random numbers or computer random number generators
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46. Systematic Samples Decide on sample size: n
Divide frame of N individuals into groups of k individuals: k=N/n
Randomly select one individual from the 1st group
Select every kth individual thereafter
N = 64
n = 8
k = 8
First Group
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47. Stratified Samples
Divide population into two or more subgroups (called strata) according to some common characteristic
A simple random sample is selected from each subgroup, with sample sizes proportional to strata sizes
Samples from subgroups are combined into one
Population
Divided
into 4
strata
Sample
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48. Cluster Samples
Population is divided into several “clusters,” each representative of the population
A simple random sample of clusters is selected
All items in the selected clusters can be used, or items can be chosen from a cluster using another probability sampling technique
Population divided into 16 clusters.
Randomly selected clusters for sample
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49. Advantages and Disadvantages
Simple random sample and systematic sample
Simple to use
May not be a good representation of the population’s underlying characteristics
Stratified sample
Ensures representation of individuals across the entire population
Cluster sample
More cost effective
Less efficient (need larger sample to acquire the same level of precision)
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50. Evaluating Survey Worthiness
What is the purpose of the survey?
Is the survey based on a probability sample?
Coverage error – appropriate frame?
Nonresponse error – follow up
Measurement error – good questions elicit good responses
Sampling error – always exists
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51. Types of Survey Errors Coverage error or selection bias
Exists if some groups are excluded from the frame and have no chance of being selected
Nonresponse error or bias
People who do not respond may be different from those who do respond
Sampling error
Variation from sample to sample will always exist
Measurement error
Due to weaknesses in question design, respondent error, and interviewer’s effects on the respondent
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52. Types of Survey Errors Coverage error Non response error
(continued)
Coverage error
Non response error
Sampling error
Measurement error
Excluded from frame
Follow up on nonresponses
Random differences from sample to sample
Bad or leading question
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53. Developing a Sampling Distribution
Assume there is a population …
Population size N=4
Random variable, X, is age of individuals
Values of X: 18, 20, 22, 24 (years) D A C B
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54. Developing a Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
P(x) .3 .2 .1 x
A B C D
Uniform Distribution
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55. Standard Error of the Mean
Different samples of the same size from the same population will yield different sample means
A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean (i.e. the standard deviation):
(This assumes that sampling is with replacement or
sampling is without replacement from an infinite population)
Note that the standard error of the mean decreases as the sample size increases
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56. Now consider all possible samples of size n=2
Developing a Sampling Distribution
(continued)
Now consider all possible samples of size n=2
16 Sample Means
1st
Obs
2nd Observation 18 20 22 24
18,18
18,20
18,22
18,24
20,18
20,20
20,22
20,24
22,18
22,20
22,22
22,24
24,18
24,20
24,22
24,24
16 possible samples (sampling with replacement)
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57. Sampling Distribution of All Sample Means
Developing a
Sampling Distribution
(continued)
Sampling Distribution of All Sample Means
Sample Means Distribution
16 Sample Means _
P(X) .3 .2 .1 _ X
(no longer uniform)
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58. Summary Measures of this Sampling Distribution:
Developing a
Sampling Distribution
(continued)
Summary Measures of this Sampling Distribution:
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59. Chapter Summary Introduced sampling distributions
Described the sampling distribution of the mean
For normal populations
Using the Central Limit Theorem
Described the sampling distribution of a proportion
Calculated probabilities using sampling distributions
Described different types of samples and sampling techniques
Examined survey worthiness and types of survey errors
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