Statistics

1
Statistics Data Analysis

• Course Number B01.1305
• Course Section 31
• Meeting Time Wednesday 6-850 pm

CLASS 5
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

3
Review of Last Class

• Special Distributions
• Counting problems
• Binomial distribution problems
• Normal distribution problems

4
CHAPTER 6

• Random Sampling and Sampling Distributions

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

6
A Scenario

• Its 900 AM on Wednesday and your boss sent you
  and email asking how your firms 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???

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, weve learned how to describe something
  that has already occurred or evaluate something
  that might occur

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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

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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

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

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Why Sample???

• We often need to know something about a large
  population.
• What is the average income of all Stern
  students?
• Its 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?

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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 gt 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

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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 dont 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

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
• Dont forget to include the college students!!!

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

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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

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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

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Choosing a Random Sample

• Number every member in the population 1N
• Use a random process to select the sample
• R, flipping a coin, random number tablewhatever
  is appropriate
• In this class we will use the computer

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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 problemno?)
• 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

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Definition
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.

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

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Example

• Suppose n200 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

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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.

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Example

• Its OK if each beer isnt exactly 12 oz so long
  as the average volume isnt 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 n24 beers

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Central Limit Theorem

• For any population, the sampling distribution of
  the sample mean is approximately normal if the
  sample size is sufficiently large

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
• datanumeric(0)
• for (i in 11000) datai mean( rexp(4) )
• hist(data)
• What type of process might follow this
  distribution???

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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?

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

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R Examples

• data rnorm(1000) do not worry about the r
  commands
• hist(data)
• qqnorm(data)
• qqline(data)
• data rexp(1000)
• hist(data)
• qqnorm(data)
• qqline(data)
• data 1-rlnorm(1000)30
• hist(data)
• qqnorm(data)
• qqline(data)
• data rnorm(1000) data15 data27
• hist(data)
• qqnorm(data)
• qqline(data)

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Point and Interval Estimation

• Chapter 7

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

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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

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

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

36
Technical Definitions
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 11000)
• data rnorm(30, mean50, sd10)
• data.meani mean(data)
• data.mediani median(data)
• Do these statistics appear unbiased?
• Which is more efficient?

38
Expressing Uncertainty
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 - a.
• 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?

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 s 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?

41
Why does it work?
42
Using the Empirical Rule Assuming Normality
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

44
Confidence Interval
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

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 cant 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
• Orthe interval is the result of a process that
  in the long run has a XX probability of being
  correct

47
Imagine Many Samples
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
• Howeverwith this added source of random errors,
  we need to handle this problem using the
  t-distribution (later on)

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?

50
Example

• Suppose that in a sample of 2,200 households with
  one or more television sets, 471 watch a
  particular networks show at a given time.
• Find a 95 confidence interval for the population
  proportion of households watching this show.

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?

52
Example

• A survey conducted found that out of 800 people,
  46 thought that Clintons 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

53
Choosing a Sample Size

• Gathering information for a statistical study can
  be expensive, time consuming, etc.
• Sothe 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

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)

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.

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 magazines 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?

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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