Statistics & Data Analysis
Course Number B01.1305
Course Section 31
Meeting Time Wednesday 6-8:50 pm
CLASS #5
Professor S. D. Balkin -- Feb. 26, 2003
- 2 -
Class #5 Outline
Understand random sampling and systematic bias Derive theoretical distribution of summary statistics Understand the Central Limit Theorem Use a normal probability plot to assess normality
Professor S. D. Balkin -- Feb. 26, 2003
- 3 -
Review of Last Class
Special Distributions• Counting problems• Binomial distribution problems• Normal distribution problems
CHAPTER 6
Random Sampling and Sampling Distributions
Professor S. D. Balkin -- Feb. 26, 2003
- 5 -
Chapter Goals
Explain why in many situations a sample is the only way learn something about a population
Explain the various methods of selecting a sample
Define and construct sampling distribution of sample means
Understand sources of bias or under-representation in data
Professor S. D. Balkin -- Feb. 26, 2003
- 6 -
A Scenario
Its 9:00 AM on Wednesday and your boss sent you and email asking how your firm’s customers would react to a new price discounting program• Your report is due tomorrow• It takes 10 minutes to interview a single customer in your database of
almost 2,000• What will you do????
Draw a sample of the customers• How will you draw the sample?• Need a representative sample
• Does your database hold a representative sample???
Professor S. D. Balkin -- Feb. 26, 2003
- 7 -
Background
Some previous chapters emphasized methods for describing data• Created frequency distributions, computed averages and measures of
dispersion
Started to lay foundation for inference by studying probability• Counting, Binomial, and Normal Distributions
• Probability distributions encompass all possible outcomes of an experiment and the probability associated with each outcome
So far, we’ve learned how to describe something that has already occurred or evaluate something that might occur
Professor S. D. Balkin -- Feb. 26, 2003
- 8 -
How are these similar…
QC department needs to check the tensile strength of steel wire• Five small pieces are selected every 5 hours• Tensile strength of each piece is determined
Marketing needs to determine the sales potential of a new drug named HappyPill. • 452 consumers were asked to try it for a week• Each consumer completed a questionnaire
Polling agency selections 2,000 voters at random and asked their approval rating of the President
In the study of insider trading, 25 CEOs were identified by the SEC and their trades were monitored for three years
Professor S. D. Balkin -- Feb. 26, 2003
- 9 -
Why Sample???
Destructive nature of some tests
Physical Impossibility of checking all items
Cost of studying all items
Adequacy of sample results
Contacting whole population would be too time-consuming
Professor S. D. Balkin -- Feb. 26, 2003
- 10 -
Types of Samples
Cross-sectional: samples are taken from an underlying population at a particular time
Time-series: samples are taken over time from a random process
Enumerative Studies: sampling from a well-defined population
Analytic Studies: look at the results of a random process to predict future behavior
Professor S. D. Balkin -- Feb. 26, 2003
- 11 -
Why Sample???
We often need to know something about a large population.• What is the average income of all Stern students?
It’s often too expensive and time-consuming to examine the entire population
Solution: Choose a small random sample and use the methods of statistical inference to draw conclusions about the population
Sampling lets us dramatically cut the costs of gathering information, but requires care. We need to ensure that the sample is representative of the population of interest
But how can any small sample be completely representative?
Professor S. D. Balkin -- Feb. 26, 2003
- 12 -
Why Sample (cont.)
IT IS IMPORTANT TO REALIZE THAT SOME INFORMATION IS LOST IF WE ONLY EXAMINE A SAMPLE OF THE ENTIRE POPULATION
Why not just use the sample mean in place of μ? For example, suppose that the average income of 100
randomly selected Stern students was = 62,154• Can we conclude that the average income of ALL Stern
students (μ) is 62,154? • Can we conclude that μ > 60,000?
Fortunately, we can use probability theory to understand how the process of taking a random sample will blur the information in a population
But first, we need to understand why and how the information is blurred
Professor S. D. Balkin -- Feb. 26, 2003
- 13 -
Sampling Variability
Although the average income of all Stern Langone students is a fixed number, the average of a sample of 100 students depends on precisely which sample is taken. In other words, the sample mean is subject to “sampling variability”
The problem is that by reporting sample mean alone, we don’t take account of the variability caused by the sampling procedure. If we had polled different students, we might have gotten a different average income
It would be a serious mistake to ignore this sampling variability, and simply assume that the mean income of all students is the same as the average of the 100 incomes given in the sample
Professor S. D. Balkin -- Feb. 26, 2003
- 14 -
Populations and Samples
You are considering opening an Atomic Wings in Bethlehem, PA• POPULATION: All residents• SAMPLE:
• Every 35th person at the mall• Every 2,000th person in the phone book• Every person who leaves Burger King• Don’t forget to include the college students!!!
Professor S. D. Balkin -- Feb. 26, 2003
- 15 -
Choosing a Representative Sample
REPRESENTATIVE: Each characteristic occurs in the same percentage of the time in the sample as in the population
BIAS: Not representative• Bias will exist if there is a systematic tendency to over/under represent
some part of the population
By deliberately not sampling based on any specific characteristic, a randomly selected sample will typically be free from bias
Randomly selecting subjects lets you make probability statements about the results
Professor S. D. Balkin -- Feb. 26, 2003
- 16 -
Examples of Bias
Selection Bias: • A telephone survey of households conducted entirely between 9 a.m.
to 5 p.m.• Using a customer complaint database to query on the new discount
program
Nonresponse Bias: Sample member refuses to participate• Every market research program
Operational Definitions: Guiding a response• Do you agree that taxes are too high in New York
Professor S. D. Balkin -- Feb. 26, 2003
- 17 -
Simple Random Sampling
Process where each possible sample of a given size has the same probability of being selected
Example: IBM reported sales of $64.792 Billion and a net loss of $2.827 Billion for 1991.• The number of individual transactions was enormous• The auditors used statistics because to choose a representative
sample of transactions to check in detail
Professor S. D. Balkin -- Feb. 26, 2003
- 18 -
Choosing a Random Sample
1. Number every member in the population 1…N
2. Use a random process to select the sample R, flipping a coin, random number table…whatever is appropriate In this class we will use the computer
Professor S. D. Balkin -- Feb. 26, 2003
- 19 -
Sampling Statistics and Distributions
Once a sample is drawn, we summarize it with sample statistics
The value of any summary statistic will vary from sample to sample (a big problem…no?)
A sample statistic is itself a random variable• Hence, it has a theoretical probability distribution called the sampling
distribution
We can find the mean and standard deviation of many random samples
Professor S. D. Balkin -- Feb. 26, 2003
- 20 -
Definition
nn
n
n
nYE
Y
)(
are Ymean sample theoferror standard and valueexpected the
,population a fromdrawn isn size of sample random a If
Professor S. D. Balkin -- Feb. 26, 2003
- 21 -
Example
Suppose the long-run average of the number of Medicare claims submitted per week to a regional office is 62,000, and the standard deviation is 7,000. • If we assume that the weekly claims submissions during a 4-week
period constitute a random sample of size 4, what are the expected value and standard error of the average weekly number of claims over a 4-week period?
NOTE: Standard error denotes the theoretically derived standard deviation of the sampling distribution of a statistic.
Professor S. D. Balkin -- Feb. 26, 2003
- 22 -
Standard Error
Standard Deviation of the statistic
Is interpreted just as you would any standard deviation
Indicates approximately how far the observed value of the statistic is from its mean• Literally: it indicated the standard deviation you would find if you took
a very large number of samples, found the sample average for each one, and worked with these sample averages as a data set
Professor S. D. Balkin -- Feb. 26, 2003
- 23 -
Example
Suppose n=200 randomly selected shoppers interviewed in a mall say they plan to spend on an average of $19.42 today with a standard deviation of $8.63• This tells you what shoppers typically plan to spend, and that a typical,
individual shopper plans to spend about $8.63 more or less than this amount
• So far, this is no more that a description of the individuals interviewed
We can say something about the unknown population mean, which is the mean amount that all shoppers in the mall today plan to spend, including those not interviewed.
What is the standard error of the mean?• This tells us the variability when we use the sample average of $19.42,
as an estimate of the unknown population mean
Professor S. D. Balkin -- Feb. 26, 2003
- 24 -
Sampling Distributions for Means and Sums
If a population distribution is Normal, then the sampling distribution of sample means is also Normal
Example: A timber company is planning to harvest 400 trees from a very large stand.• Yield is determined by its diameter• Distribution of diameters is normal with mean 44 inches and standard
deviation of 4 inches• Find the probability that the average diameter of the harvest trees is
between 43.5 and 44.5 inches.
Professor S. D. Balkin -- Feb. 26, 2003
- 25 -
Example
Its OK if each beer isn’t exactly 12 oz so long as the average volume isn’t too low or too high.• In your production facility, you know that the volume of each beer
follows a Normal distribution, has a standard deviation of 0.5 ounces, representing variability about their mean of 12.01 oz.
• Any case (24 beers) that has an average weight per beer less than 11.75 ounces will be rejected.
What fraction of cases will be rejected this way?• First find the mean and standard deviation of the average of n=24
beers
Professor S. D. Balkin -- Feb. 26, 2003
- 26 -
Central Limit Theorem
For any population, the sampling distribution of the sample mean is approximately normal if the sample size is sufficiently large
Professor S. D. Balkin -- Feb. 26, 2003
- 27 -
Simulation Example
Use R to draw 1000 samples each, with sample sizes 4, 10, 30, and 60 from a highly right-skewed distribution having mean and standard deviation both equal to 1.
Display a histogram of the sample means
data=numeric(0)
for (i in 1:1000) data[i] = mean( rexp(4) )
hist(data)
What type of process might follow this distribution???
Professor S. D. Balkin -- Feb. 26, 2003
- 28 -
Example of Use
An agency of the Commerce Department in a certain state wishes to check the accuracy of weights in supermarkets
They decide to weigh 9 packages of ground meat labeled as 1 pound packages
They will investigate any supermarket where the average weight of the packages is less than 15.5 oz
Assuming that the standard deviation of package weights is 0.6 oz, what is the probability they will investigate an honest market?
Professor S. D. Balkin -- Feb. 26, 2003
- 29 -
Normal Probability Plot
Plots actual versus expected values, assuming a normal distribution
• Nearly normal data will plot as a near straight line
• Right-skewed data plot as a curve, with the slope getting steeper as one moves to the right
• Left-skewed data plot as a curve, with the slope getting flatter as one moves to the right
• Symmetric but outlier-prone data plot as an S-shape, with the slope steepest at both sides
Professor S. D. Balkin -- Feb. 26, 2003
- 30 -
R Examples
data = rnorm(1000) ## do not worry about the r*** commandshist(data)qqnorm(data)qqline(data)
data = rexp(1000)hist(data)qqnorm(data)qqline(data)
data = 1-rlnorm(1000)+30hist(data)qqnorm(data)qqline(data)
data = rnorm(1000); data[1]=5; data[2]=7;hist(data)qqnorm(data)qqline(data)
Point and Interval Estimation
Chapter 7
Professor S. D. Balkin -- Feb. 26, 2003
- 32 -
Review
Basic problem of statistical theory is how to infer a population or process value given only sample data
Any sample statistic will vary from sample to sample
Any sample statistic will differ from the true, population value
Must consider random error in sample statistic estimation
Professor S. D. Balkin -- Feb. 26, 2003
- 33 -
Chapter Goals
Summarize sample data• Choosing an estimator
• Unbiased estimator
Constructing confidence intervals for means with known standard deviation
Constructing confidence intervals for proportions
Determining how large a sample is needed
Constructing confidence intervals when standard deviation is not known
Understanding key underlying assumptions underlying confidence interval methods
Professor S. D. Balkin -- Feb. 26, 2003
- 34 -
Reminder: Statistical Inference
Problem of Inferential Statistics:• Make inferences about one or more population parameters based on
observable sample data
Forms of Inference:• Point estimation: single best guess regarding a population parameter• Interval estimation: Specifies a reasonable range for the value of the
parameter• Hypothesis testing: Isolating a particular possible value for the
parameter and testing if this value is plausible given the available data
Professor S. D. Balkin -- Feb. 26, 2003
- 35 -
Point Estimators
Computing a single statistic from the sample data to estimate a population parameter
Choosing a point estimator:• What is the shape of the distribution?• Do you suspect outliers exist?• Plausible choices:
• Mean
• Median
• Mode
• Trimmed Mean
Professor S. D. Balkin -- Feb. 26, 2003
- 36 -
Technical Definitions
estimators unbiased
possible all oferror standardsmallest thehasit if problem particular afor
efficientmost called isestimator An :ESTIMATOR EFFICIENT
equals valueexpected its if
parameter population for the unbiased called is data sample theof
function a is that ˆestimator An :ESTIMATOR UNBIASED
on.distributi sampling al theoretica hasit
thereforeand variablerandom a itself isestimator An .for
estimatepoint a yields that sample random a offunction
a is parameter a of ˆestimator An :ESTIMATOR
Professor S. D. Balkin -- Feb. 26, 2003
- 37 -
Example
I used R to draw 1,000 samples, each of size 30, from a normally distributed population having mean 50 and standard deviation 10.
For each sample the mean and median are computed.
data.mean = numeric(0)
data.median = numeric(0)
for(i in 1:1000) {
data = rnorm(30, mean=50, sd=10)
data.mean[i] = mean(data)
data.median[i] = median(data)
}
Do these statistics appear unbiased?
Which is more efficient?
Professor S. D. Balkin -- Feb. 26, 2003
- 38 -
Expressing Uncertainty
accuracy. complete with estimates that impression
false theleavemay alone of reporting thee,Furthermor
y.reliabilitown itsabout n informatio
no containsit because usefulness limited of is itself,by
Used.parameter theofestimator point a is mean sample The
. size of sample aon based mean
population aabout inferences make to tryingare weSuppose
X
X
X
X
n
Professor S. D. Balkin -- Feb. 26, 2003
- 39 -
Confidence Interval
An interval with random endpoints which contains the parameter of interest (in this case, μ) with a pre-specified probability, denoted by 1 - α.
The confidence interval automatically provides a margin of error to account for the sampling variability of the sample statistic.
Example: A machine is supposed to fill “12 ounce” bottles of Guinness. To see if the machine is working properly, we randomly select 100 bottles recently filled by the machine, and find that the average amount of Guinness is 11.95 ounces. Can we conclude that the machine is not working properly?
Professor S. D. Balkin -- Feb. 26, 2003
- 40 -
No! By simply reporting the sample mean, we are neglecting the fact that the amount of beer varies from bottle to bottle and that the value of the sample mean depends on the luck of the draw
It is possible that a value as low as 11.75 is within the range of natural variability for the sample mean, even if the average amount for all bottles is in fact μ = 12 ounces.
Suppose we know from past experience that the amounts of beer in bottles filled by the machine have a standard deviation of σ = 0.05 ounces.
Since n = 100, we can assume (using the Central Limit Theorem) that the sample mean is normally distributed with mean μ (unknown) and standard error 0.005
What does the Empirical Rule tell us about the average volume of the sample mean?
Professor S. D. Balkin -- Feb. 26, 2003
- 41 -
Why does it work?
X
time theof 95%
here in is XXS
time theof 95%
about here in is
X
Professor S. D. Balkin -- Feb. 26, 2003
- 42 -
Using the Empirical Rule Assuming Normality
Professor S. D. Balkin -- Feb. 26, 2003
- 43 -
Confidence Intervals
“Statistics is never having to say you're certain”.• (Tee shirt, American Statistical Association).
Any sample statistic will vary from sample to sample Point estimates are almost inevitably in error to some
degree Thus, we need to specify a probable range or interval
estimate for the parameter
Professor S. D. Balkin -- Feb. 26, 2003
- 44 -
Confidence Interval
YY zyzy
2/2/
:mean sample theoferror standard
the times valuetable-z a toequal termminus-or-plus a error with sampling
for allow mean, population theof estimatean asmean sample theUsing
KNOWN AND FOR INTERVAL CONFIDENCE )%1(100
Professor S. D. Balkin -- Feb. 26, 2003
- 45 -
Example
An airline needs an estimate of the average number of passengers on a newly scheduled flight
Its experience is that data for the first month of flights are unreliable, but thereafter the passenger load settles down
The mean passenger load is calculated for the first 20 weekdays of the second month after initiation of this particular flight
If the sample mean is 112 and the population standard deviation is assumed to be 25, find a 90% confidence interval for the true, long-run average number of passengers on this flight
Professor S. D. Balkin -- Feb. 26, 2003
- 46 -
Interpretation
The significance level of the confidence interval refers to the process of constructing confidence intervals
Each particular confidence interval either does or does not include the true value of the parameter being estimated
We can’t say that this particular estimate is correct to within the error
So, we say that we have a XX% confidence that the population parameter is contained in the interval
Or…the interval is the result of a process that in the long run has a XX% probability of being correct
Professor S. D. Balkin -- Feb. 26, 2003
- 47 -
Imagine Many Samples
22 23 24
The interval you computed
Missed!Missed!
The population mean = 23.29
Other intervals y
ou
might have computed
Professor S. D. Balkin -- Feb. 26, 2003
- 48 -
Getting Realistic
The population standard deviation is rarely known Usually both the mean and standard deviation must be
estimated from the sample Estimate with s However…with this added source of random errors, we need
to handle this problem using the t-distribution (later on)
Professor S. D. Balkin -- Feb. 26, 2003
- 49 -
Confidence Intervals for Proportions
We can also construct confidence intervals for proportions of successes
Recall that the expected value and standard error for the number of successes in a sample are:
How can we construct a confidence interval for a proportion?
nE /)1(;)ˆ( ˆ
Professor S. D. Balkin -- Feb. 26, 2003
- 50 -
Example
Suppose that in a sample of 2,200 households with one or more television sets, 471 watch a particular network’s show at a given time.
Find a 95% confidence interval for the population proportion of households watching this show.
Professor S. D. Balkin -- Feb. 26, 2003
- 51 -
Example
The 1992 presidential election looked like a very close three-way race at the time when news polls reported that of 1,105 registered voters surveyed:• Perot: 33%• Bush: 31%• Clinton: 28%
Construct a 95% confidence interval for Perot? What is the margin of error? What happened here?
Professor S. D. Balkin -- Feb. 26, 2003
- 52 -
Example
A survey conducted found that out of 800 people, 46% thought that Clinton’s first approved budget represented a major change in the direction of the country.
Another 45% thought it did not represent a major change. Compute a 95% confidence interval for the percent of people
who had a positive response. What is the margin of error? Interpret…
Professor S. D. Balkin -- Feb. 26, 2003
- 53 -
Choosing a Sample Size
Gathering information for a statistical study can be expensive, time consuming, etc.
So…the question of how much information to gather is very important
When considering a confidence interval for a population mean , there are three quantities to consider:
n
z
Y /
2/
Professor S. D. Balkin -- Feb. 26, 2003
- 54 -
Choosing a Sample Size (cont)
Tolerability Width: The margin of acceptable error 3% $10,000
Derive the required sample size using:• Margin of error (tolerability width)• Level of Significance (z-value)• Standard deviation (given, assumed, or calculated)
Professor S. D. Balkin -- Feb. 26, 2003
- 55 -
Example
Union officials are concerned about reports of inferior wages being paid to employees of a company under its jurisdiction
How large a sample is needs to obtain a 90% confidence interval for the population mean hourly wage with width equal to $1.00? Assume that =4.
Professor S. D. Balkin -- Feb. 26, 2003
- 56 -
Example
A direct-mail company must determine its credit policies very carefully. The firm suspects that advertisements in a certain magazine have led to
an excessively high rate of write-offs. The firm wants to establish a 90% confidence interval for this magazine’s
write-off proportion that is accurate to 2.0%• How many accounts must be sampled to guarantee this goal?• If this many accounts are sampled and 10% of the sampled accounts are
determined to be write-offs, what is the resulting 90% confidence interval?• What kind of difference do we see by using an observed proportion over a
conservative guess?
Professor S. D. Balkin -- Feb. 26, 2003
- 57 -
Homework #5
Hildebrand/Ott• 6.4• 6.5• 6.8• 6.16• 6.17• 6.46
• In (a) create a normal probability plot also and interpret
• 7.1• 7.2• 7.14• 7.17• 7.18• 7.20• 7.21• 7.30• Read Chapter 11
Verzani
Top Related