Chapter 4
Probability and Sampling
Distributions
1
Random Variable


Definition: A random variable is a variable
whose value is a numerical outcome of a
random phenomenon.
...
Section 4.4
The Sampling Distribution of a
Sample Mean
3
Introduction
A statistic from a random sample will take
different values if we take more samples
from the same population
...
Example



Suppose that we are interested in the workout
times of ISU students at the Recreation center.
Let’s assume th...
Example


A SRS should be a fairly good representation of
the population so the x-bar should be somewhere
near the µ.
fro...
Statistical Estimation


If x-bar is rarely exactly right and varies from
sample to sample, why is x-bar a reasonable
est...
The Law of Large Numbers


Law of Large Numbers (LLN):
 Draw

independent observations at random from
any population wit...
Example


Suppose we have a bowl with 21 small pieces of paper
inside. Each paper is labeled with a number 0-20. We
will ...
Example


Draw a second random sample of size n = 5. Calculate x for this
sample.

x



Draw a third random sample of si...
Another Example

5.710
5.705
5.700
5.695

Mean of first n observations



Example: Suppose we know that the average heigh...
Example 4.21 From Book




Sulfur compounds such as dimethyl sulfide (DMS)
are sometimes present in wine
DMS causes “off...
Example 4.21 From Text




To estimate µ, we present tasters with both
natural wine and the same wine spiked with DMS
at...
Example
Suppose µ = 25 is the true value of the
parameter we seek to estimate
 The first subject had threshold 28 so the
...
Example 4.21From Book
The law of large
numbers in action: as
we take more
observations, the
sample mean x
always approache...
The Law of Large Numbers


The law of large numbers is the foundation of
business enterprises such as casinos and
insuran...
Thinking about the
Law of Large Numbers
The Law of Large Numbers says broadly that
the average results of many independent...
The “Law of Small Numbers”
or “Averages”


The Law of Large Numbers describes the
regular behavior of chance phenomena in...
The “Law of Small Numbers”
or “Averages”


Example: Pretend you have an average free throw
success rate of 70%. One day o...
The Hot Hand Debate







In some sports If player makes several consecutive
good plays, like a few good golf shots i...
The Gambling Hot Hand



Gamblers often follow the hot-hand theory,
betting that a “lucky” run will continue
At other ti...
Need for Law of Large Numbers


Our inability to accurately distinguish
random behavior from systematic
influences points...
How Large is a Large
Number?


The Law of Large Numbers says that the actual
mean outcome of many trials gets close to th...
More Laws of Large Numbers


The Law of Large Numbers is one of the central
facts about probability
 LLN

explains why g...
What if Observations are not
Independent








You are in charge of a process that
manufactures video screens for co...
AYK 4.82


Use the Law of Large Numbers applet on
the text book website

26
Sampling Distributions


The Law of Large Numbers assures us that if
we measure enough subjects, the statistic xbar will ...
Sampling Distributions


What if we don’t have a large sample?
 Take

a large number of samples of the same
size from th...


The idea of a sampling distribution is
the foundation of statistical inference
 The

laws of probability can tell us a...
Mean and Standard Deviation of a
Sample Mean


Suppose that x-bar is the mean of a SRS of size
n drawn from a large popul...
Mean and Standard Deviation of a
Sample Mean


The mean of the statistic x-bar is always the same
as the mean µ of the po...
Mean and Standard Deviation of a
Sample Mean


An unbiased estimator is “correct on the
average” in many samples
 how

c...
Mean and Standard Deviation of a
Sample Mean


Not only is the standard deviation of the
distribution of x-bar smaller th...
Mean and Standard Deviation of a
Sample Mean


If n is large, the standard deviation of x-bar is
small and almost all sam...
Example


Suppose we take samples of size 15
from a distribution with mean 25 and
standard deviation 7
 the

distributio...
What About Shape?


We have described the center and spread
of the sampling distribution of a sample
mean x-bar, but not ...
Sampling Distribution of a
Sample Mean


If a population has the N(µ, σ) distribution,
then the sample mean x-bar of n
in...
Example


Adults differ in the smallest amount of
dimethyl sulfide they can detect in wine



Extensive studies have fou...
Example


Because the population distribution is
Normal, the sampling distribution of xbar is also Normal



If n = 10, ...
What if the Population Distribution
is not Normal?


As the sample size increases, the distribution
of x-bar changes shap...
Central Limit Theorem


Draw a SRS of size n from any population
with mean µ and finite standard deviation σ



When n i...
Central Limit Theorem


More general versions of the central limit
theorem say that the distribution of a sum or
average ...
How Large a Sample is Needed?


Sample Size depends on whether the
population distribution is close to Normal
 We

requi...
Example





The time X that a technician requires to perform
preventive maintenance on an air-conditioning unit
is gov...
Example


What is the probability that P ( x > 0.83)
your company’s units


average maintenance time


 x − µ > 0.83...
4.86 ACT scores


The scores of students on the ACT college
entrance examination in a recent year had
the Normal distribu...
4.86 ACT scores


What is the probability that a single
student randomly chosen from all those
taking the test scores 21 ...
4.86 ACT scores


About 34% of students (from this
population) scored a 21 or higher on the
ACT



The probability that ...
4.86 ACT scores


Now take a SRS of 50 students who took
the test. What are the mean and standard
deviation of the sample...
4.86 ACT scores


What is the probability that the mean
score x-bar of these students is 21 or
higher?
P ( x ≥ 21)



...
4.86 ACT scores


About 0.2 % of all random samples of size
50 (from this population) would have a
mean score x-bar of 21...
Section 4.4 Summary


When we want information about the
population mean µ for some variable, we
often take a SRS and use...
Section 4.4 Summary


The Law of Large Numbers states that
the actually observed mean outcome xbar must approach the mean...
Section 4.4 Summary


The sampling distribution of x-bar
describes how the statistic x-bar varies in
all possible samples...
Section 4.4 Summary


The mean of the sampling distribution is
µ, so that x-bar is an unbiased estimator
of µ.

55
Section 4.4 Summary


The standard deviation of the sampling
distribution of x-bar is sigma over the
square root of n for...
Section 4.4 Summary


If the population has a Normal distribution,
so does x-bar.

57
Section 4.4 Summary


The Central Limit Theorem states that for large
n the sampling distribution of x-bar is
approximate...
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  1. 1. Chapter 4 Probability and Sampling Distributions 1
  2. 2. Random Variable  Definition: A random variable is a variable whose value is a numerical outcome of a random phenomenon.  The statistic calculated from a randomly chosen sample is an example of a random variable.  We don’t know the exact outcome beforehand. A statistic from a random sample will take different values if we take more samples from the same population. 2
  3. 3. Section 4.4 The Sampling Distribution of a Sample Mean 3
  4. 4. Introduction A statistic from a random sample will take different values if we take more samples from the same population  The values of a statistic do no vary haphazardly from sample to sample but have a regular pattern in many samples   We  already saw the sampling distribution We’re going to discuss an important sampling distribution. The sampling distribution of the sample mean,4x-bar( )
  5. 5. Example   Suppose that we are interested in the workout times of ISU students at the Recreation center. Let’s assume that μ is the average workout time of all ISU students  To estimate μ lets take a simple random sample of 100 students at ISU   We will record each students work out time (x) Then we find the average workout time for the 100 students    x The population mean μ is the parameter of interest. The sample mean, x , is the statistic (which is a random variable). Use x to estimate μ (This seems like a sensible thing to do). 5
  6. 6. Example  A SRS should be a fairly good representation of the population so the x-bar should be somewhere near the µ. from a SRS is an unbiased estimate of µ due to the randomization  x-bar  We don’t expect x-bar to be exactly equal to µ  There  is variability in x-bar from sample to sample If we take another simple random sample (SRS) of 100 students, then the x-bar will probably be different.  Why, then, can I use the results of one sample to estimate µ? 6
  7. 7. Statistical Estimation  If x-bar is rarely exactly right and varies from sample to sample, why is x-bar a reasonable estimate of the population mean µ?  Answer: if we keep on taking larger and larger samples, the statistic x-bar is guaranteed to get closer and closer to the parameter µ  We have the comfort of knowing that if we can afford to keep on measuring more subjects, eventually we will estimate the mean amount of workout time for ISU students very accurately 7
  8. 8. The Law of Large Numbers  Law of Large Numbers (LLN):  Draw independent observations at random from any population with finite mean µ  As the number of observations drawn increases, the mean x-bar of the observed values gets closer and closer to the mean µ of the population  If n is the sample size as n gets large x → µ  The Law of Large Numbers holds for any population, not just for special classes such as Normal distributions 8
  9. 9. Example  Suppose we have a bowl with 21 small pieces of paper inside. Each paper is labeled with a number 0-20. We will draw several random samples out of the bowl of size n and record the sample means, x-bar for each sample.    What is the population? Since we know the values for each individual in the population (i.e. for each paper in the bowl), we can actually calculate the value of µ, the true population mean. µ = 10 Draw a random sample of size n = 1. Calculate x-bar for this sample. 9
  10. 10. Example  Draw a second random sample of size n = 5. Calculate x for this sample. x  Draw a third random sample of size n = 10. Calculate sample.  Draw a fourth random sample of size n = 15. Calculate sample.  Draw a fifth random sample of size n = 20. Calculate sample.  What can we conclude about the value of increases? for this x for this x for this x as the sample size THIS IS CALLED THE LAW OF LARGE NUMBERS. 10
  11. 11. Another Example 5.710 5.705 5.700 5.695 Mean of first n observations  Example: Suppose we know that the average height of all high school students in Iowa is 5.70 feet. We get SRS’s from the population and calculate the height. mean of first n observations (feet)  0 5000 10000 number of observations 15000 11 20000
  12. 12. Example 4.21 From Book   Sulfur compounds such as dimethyl sulfide (DMS) are sometimes present in wine DMS causes “off-odors” in wine, so winemakers want to know the odor threshold  What is the lowest concentration of DMS that the human nose can detect  Different people have different thresholds, so we start by asking about the mean threshold µ in the population of all adults µ is a parameter that describes this population 12
  13. 13. Example 4.21 From Text   To estimate µ, we present tasters with both natural wine and the same wine spiked with DMS at different concentrations to find the lowest concentration at which they can identify the spiked wine The odor thresholds for 10 randomly chosen subjects (in micrograms/liter):  28  40 28 33 20 31 29 27 17 21 The mean threshold for these subjects is 27.4  x-bar is a statistic calculated from this sample  A statistic, such as the mean of a random sample of 13 10 adults, is a random variable.
  14. 14. Example Suppose µ = 25 is the true value of the parameter we seek to estimate  The first subject had threshold 28 so the line starts there  The second point is the mean of the first 28 + 40 two subjects: x= = 34 2   This process continues many many times, and our line begins to settle around µ = 25 14
  15. 15. Example 4.21From Book The law of large numbers in action: as we take more observations, the sample mean x always approaches the mean of the population µ = 25 15
  16. 16. The Law of Large Numbers  The law of large numbers is the foundation of business enterprises such as casinos and insurance companies  The winnings (or losses) of a gambler on a few plays are uncertain -- that’s why gambling is exciting(?)  But, the “house” plays tens of thousands of times  So the house, unlike individual gamblers, can count on the long-run regularity described by the Law of Large Numbers  The average winnings of the house on tens of thousands of plays will be very close to the mean of the distribution of winnings  Hence, the LLN guarantees the house a profit! 16
  17. 17. Thinking about the Law of Large Numbers The Law of Large Numbers says broadly that the average results of many independent observations are stable and predictable  A grocery store deciding how many gallons of milk to stock and a fast-food restaurant deciding how many beef patties to prepare can predict demand even though their customers make independent decisions   The Law of Large Numbers says that the many individual decisions will produce a stable result 17
  18. 18. The “Law of Small Numbers” or “Averages”  The Law of Large Numbers describes the regular behavior of chance phenomena in the long run  Many people believe in an incorrect “law of small numbers”  We falsely expect even short sequences of random events to show the kind of average behaviors that in fact appears only in the long run 18
  19. 19. The “Law of Small Numbers” or “Averages”  Example: Pretend you have an average free throw success rate of 70%. One day on the free throw line, you miss 8 shots in a row. Should you hit the next shot by the mythical “law of averages.”  No. The law of large numbers tells us that the long run average will be close to 70%. Missing 8 shots in a row simply means you are having a bad day. 8 shots is hardly the “long run”. Furthermore, the law of large numbers says nothing about the next event. It only tells us what will happen if we keep track of the long run average. 19
  20. 20. The Hot Hand Debate     In some sports If player makes several consecutive good plays, like a few good golf shots in a row, often they claim to have the “hot hand”, which generally implies that their next shot is likely to a good one. There have been studies that suggests that runs of golf shots good or bad are no more frequent in golf than would be expected if each shot were independent of the player’s previous shots Players perform consistently, not in streaks Our perception of hot or cold streaks simply shows that we don’t perceive random behavior very well! 20
  21. 21. The Gambling Hot Hand   Gamblers often follow the hot-hand theory, betting that a “lucky” run will continue At other times, however, they draw the opposite conclusion when confronted with a run of outcomes  If a coin gives 10 straight heads, some gamblers feel that it must now produce some extra tails to get back into the average of half heads and half tails  Not true! If the next 10,000 tosses give about 50% tails, those 10 straight heads will be swamped by the later thousands of heads and tails.  No short run compensation is needed to get back to 21 the average in the long run.
  22. 22. Need for Law of Large Numbers  Our inability to accurately distinguish random behavior from systematic influences points out the need for statistical inference to supplement exploratory analysis of data  Probability calculations can help verify that what we see in the data is more than a random pattern 22
  23. 23. How Large is a Large Number?  The Law of Large Numbers says that the actual mean outcome of many trials gets close to the distribution mean µ as more trials are made  It doesn’t say how many trials are needed to guarantee a mean outcome close to µ  That  depends on the variability of the random outcomes The more variable the outcomes, the more trials are needed to ensure that the mean outcome xbar is close to the distribution µ 23
  24. 24. More Laws of Large Numbers  The Law of Large Numbers is one of the central facts about probability  LLN explains why gambling, casinos, and insurance companies make money  LLN assures us that statistical estimation will be accurate if we can afford enough observations  The basic Law of Large Numbers applies to independent observations that all have the same distribution  Mathematicians general settings have extended the law to many more 24
  25. 25. What if Observations are not Independent     You are in charge of a process that manufactures video screens for computer monitors Your equipment measures the tension on the metal mesh that lies behind each screen and is critical to its image quality You want to estimate the mean tension µ for the process by the average x-bar of the measurements The tension measurements are not independent 25
  26. 26. AYK 4.82  Use the Law of Large Numbers applet on the text book website 26
  27. 27. Sampling Distributions  The Law of Large Numbers assures us that if we measure enough subjects, the statistic xbar will eventually get very close to the unknown parameter µ 27
  28. 28. Sampling Distributions  What if we don’t have a large sample?  Take a large number of samples of the same size from the same population  Calculate  Make  the sample mean for each sample a histogram of the sample means the histogram of values of the statistic approximates the sampling distribution that we would see if we kept on sampling forever… 28
  29. 29.  The idea of a sampling distribution is the foundation of statistical inference  The laws of probability can tell us about sampling distributions without the need to actually choose or simulate a large number of samples 29
  30. 30. Mean and Standard Deviation of a Sample Mean  Suppose that x-bar is the mean of a SRS of size n drawn from a large population with mean µ and standard deviation σ  The mean of the sampling distribution of x-bar is σ µ and its standard deviation is n  Notice: averages are less variable than individual observations! 30
  31. 31. Mean and Standard Deviation of a Sample Mean  The mean of the statistic x-bar is always the same as the mean µ of the population sampling distribution of x-bar is centered at µ  in repeated sampling, x-bar will sometimes fall above the true value of the parameter µ and sometimes below, but there is no systematic tendency to overestimate or underestimate the parameter  because the mean of x-bar is equal to µ, we say that the statistic x-bar is an unbiased estimator of the parameter µ  the 31
  32. 32. Mean and Standard Deviation of a Sample Mean  An unbiased estimator is “correct on the average” in many samples  how close the estimator falls to the parameter in most samples is determined by the spread of the sampling distribution  if individual observations have standard deviation σ, then sample means x-bar from samples of size n σ have standard deviation n  Again, notice that averages are less variable than individual observations 32
  33. 33. Mean and Standard Deviation of a Sample Mean  Not only is the standard deviation of the distribution of x-bar smaller than the standard deviation of individual observations, but it gets smaller as we take larger samples  The results of large samples are less variable than the results of small samples  Remember, we divided by the square root of n 33
  34. 34. Mean and Standard Deviation of a Sample Mean  If n is large, the standard deviation of x-bar is small and almost all samples will give values of xbar that lie very close to the true parameter µ  The sample mean from a large sample can be trusted to estimate the population mean accurately  Notice, that the standard deviation of the sample distribution gets smaller only at the rate n  To cut the standard deviation of x-bar in half, we must take four times as many observations, not just twice as many (square root of 4 is 2) 34
  35. 35. Example  Suppose we take samples of size 15 from a distribution with mean 25 and standard deviation 7  the distribution of x-bar is:  the mean of x-bar is:  25  the  7    25,   15  standard deviation of x-bar is: 1.80739 35
  36. 36. What About Shape?  We have described the center and spread of the sampling distribution of a sample mean x-bar, but not its shape  The shape of the distribution of x-bar depends on the shape of the population distribution 36
  37. 37. Sampling Distribution of a Sample Mean  If a population has the N(µ, σ) distribution, then the sample mean x-bar of n independent observations has the σ  N µ ,  distribution n   37
  38. 38. Example  Adults differ in the smallest amount of dimethyl sulfide they can detect in wine  Extensive studies have found that the DMS odor threshold of adults follows roughly a Normal distribution with mean µ = 25 micrograms per liter and standard deviation σ = 7 micrograms per liter 38
  39. 39. Example  Because the population distribution is Normal, the sampling distribution of xbar is also Normal  If n = 10, what is the distribution of xbar? 7   N  25,  10   39
  40. 40. What if the Population Distribution is not Normal?  As the sample size increases, the distribution of x-bar changes shape  The distribution looks less like that of the population and more like a Normal distribution  When the sample is large enough, the distribution of x-bar is very close to Normal  This result is true no matter what shape of the population distribution as long as the population has a finite standard deviation σ 40
  41. 41. Central Limit Theorem  Draw a SRS of size n from any population with mean µ and finite standard deviation σ  When n is large, the sampling distribution of the sample mean x-bar is approximately Normal: x-bar is approximately σ  N µ ,  n   41
  42. 42. Central Limit Theorem  More general versions of the central limit theorem say that the distribution of a sum or average of many small random quantities is close to Normal  The central limit theorem suggests why the Normal distributions are common models for observed data 42
  43. 43. How Large a Sample is Needed?  Sample Size depends on whether the population distribution is close to Normal  We require more observations if the shape of the population distribution is far from Normal 43
  44. 44. Example    The time X that a technician requires to perform preventive maintenance on an air-conditioning unit is governed by the Exponential distribution (figure 4.17 (a)) with mean time µ = 1 hour and standard deviation σ = 1 hour Your company operates 70 of these units The distribution of the mean time your company spends on preventative maintenance is: 1   N 1,  = N (1,0.12 ) 70   44
  45. 45. Example  What is the probability that P ( x > 0.83) your company’s units   average maintenance time    x − µ > 0.83 − 1  exceeds 50 minutes? =P  σ  0.12  50/60 = 0.83 hour     So we want to know P(x-bar >  n   0.83) = P ( z > −1.42 )   Use Normal distribution calculations we learned in Chapter 2! = 1 − P ( z < −1.42 ) = 1 − 0.0778 = 0.9222 45
  46. 46. 4.86 ACT scores  The scores of students on the ACT college entrance examination in a recent year had the Normal distribution with mean µ = 18.6 and standard deviation σ = 5.9 46
  47. 47. 4.86 ACT scores  What is the probability that a single student randomly chosen from all those taking the test scores 21 or higher? P( x ≥ 21)  x − µ 21 − 18.6  = P ≥  5.9   σ = P ( z ≥ 0.4068) = 1 − P ( z < 0.41) = 1 − 0.6591 = 0.3409 47
  48. 48. 4.86 ACT scores  About 34% of students (from this population) scored a 21 or higher on the ACT  The probability that a single student randomly chosen from this population would have a score of 21 or higher is 0.34 48
  49. 49. 4.86 ACT scores  Now take a SRS of 50 students who took the test. What are the mean and standard deviation of the sample mean score x-bar of these 50 students?  Mean = 18.6 [same as µ]  Standard Deviation = 0.8344 [sigma/sqrt(50)] 49
  50. 50. 4.86 ACT scores  What is the probability that the mean score x-bar of these students is 21 or higher? P ( x ≥ 21)     x − µ 21 − 18.6  = P ≥  σ  0.834      n   = P ( z ≥ 2.8778) = 1 − P ( z < 2.88) = 1 − 0.9980 = 0.002 50
  51. 51. 4.86 ACT scores  About 0.2 % of all random samples of size 50 (from this population) would have a mean score x-bar of 21 or higher.  The probability of having a mean score xbar of 21 or higher from a sample of 50 students (from this population) is 0.002. 51
  52. 52. Section 4.4 Summary  When we want information about the population mean µ for some variable, we often take a SRS and use the sample mean x-bar to estimate the unknown parameter µ. 52
  53. 53. Section 4.4 Summary  The Law of Large Numbers states that the actually observed mean outcome xbar must approach the mean µ of the population as the number of observations increases. 53
  54. 54. Section 4.4 Summary  The sampling distribution of x-bar describes how the statistic x-bar varies in all possible samples of the same size from the same population. 54
  55. 55. Section 4.4 Summary  The mean of the sampling distribution is µ, so that x-bar is an unbiased estimator of µ. 55
  56. 56. Section 4.4 Summary  The standard deviation of the sampling distribution of x-bar is sigma over the square root of n for a SRS of size n if the population has standard deviation sigma. That is, averages are less variable than individual observations. 56
  57. 57. Section 4.4 Summary  If the population has a Normal distribution, so does x-bar. 57
  58. 58. Section 4.4 Summary  The Central Limit Theorem states that for large n the sampling distribution of x-bar is approximately Normal for any population with finite standard deviation sigma. That is, averages are more Normal than individual observations. We can use the fact that x-bar has a known Normal distribution to calculate approximate probabilities for events involving xbar. 58

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