Chapter 7 Statistical Inference: Confidence Intervals

1
Chapter 7Statistical Inference Confidence
Intervals

• Learn .
• How to Estimate a Population
• Parameter Using Sample Data

2
Section 7.1

• What Are Point and Interval Estimates of
  Population Parameters?

3
Point Estimate

• A point estimate is a single number that is our
  best guess for the parameter

4
Interval Estimate

• An interval estimate is an interval of numbers
  within which the parameter value is believed to
  fall.

5
Point Estimate vs Interval Estimate
6
Point Estimate vs Interval Estimate

• A point estimate doesnt tell us how close the
  estimate is likely to be to the parameter
• An interval estimate is more useful
• It incorporates a margin of error which helps us
  to gauge the accuracy of the point estimate

7
Point Estimation How Do We Make a Best Guess
for a Population Parameter?

• Use an appropriate sample statistic
• For the population mean, use the sample mean
• For the population proportion, use the sample
  proportion

8
Point Estimation How Do We Make a Best Guess
for a Population Parameter?

• Point estimates are the most common form of
  inference reported by the mass media

9
Properties of Point Estimators

• Property 1 A good estimator has a sampling
  distribution that is centered at the parameter
• An estimator with this property is unbiased
• The sample mean is an unbiased estimator of the
  population mean
• The sample proportion is an unbiased estimator of
  the population proportion

10
Properties of Point Estimators

• Property 2 A good estimator has a small
  standard error compared to other estimators
• This means it tends to fall closer than other
  estimates to the parameter

11
Interval Estimation Constructing an Interval
that Contains the Parameter (We Hope!)

• Inference about a parameter should provide not
  only a point estimate but should also indicate
  its likely precision

12
Confidence Interval

• A confidence interval is an interval containing
  the most believable values for a parameter
• The probability that this method produces an
  interval that contains the parameter is called
  the confidence level
• This is a number chosen to be close to 1, most
  commonly 0.95

13
What is the Logic Behind Constructing a
Confidence Interval?

• To construct a confidence interval for a
  population proportion, start with the sampling
  distribution of a sample proportion

14
The Sampling Distribution of the Sample Proportion

• Gives the possible values for the sample
  proportion and their probabilities
• Is approximately a normal distribution for large
  random samples
• Has a mean equal to the population proportion
• Has a standard deviation called the standard
  error

15
A 95 Confidence Interval for a Population
Proportion

• Fact Approximately 95 of a normal distribution
  falls within 1.96 standard deviations of the mean
• That means With probability 0.95, the sample
  proportion falls within about 1.96 standard
  errors of the population proportion

16
Margin of Error

• The margin of error measures how accurate the
  point estimate is likely to be in estimating a
  parameter
• The distance of 1.96 standard errors in the
  margin of error for a 95 confidence interval

17
Confidence Interval

• A confidence interval is constructed by adding
  and subtracting a margin of error from a given
  point estimate
• When the sampling distribution is approximately
  normal, a 95 confidence interval has margin of
  error equal to 1.96 standard errors

18
Section 7.2

• How Can We Construct a Confidence Interval to
  Estimate a Population Proportion?

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Finding the 95 Confidence Interval for a
Population Proportion

• We symbolize a population proportion by p
• The point estimate of the population proportion
  is the sample proportion
• We symbolize the sample proportion by

20
Finding the 95 Confidence Interval for a
Population Proportion

• A 95 confidence interval uses a margin of error
  1.96(standard errors)
• point estimate margin of error

21
Finding the 95 Confidence Interval for a
Population Proportion

• The exact standard error of a sample proportion
  equals
• This formula depends on the unknown population
  proportion, p
• In practice, we dont know p, and we need to
  estimate the standard error

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Finding the 95 Confidence Interval for a
Population Proportion

• In practice, we use an estimated standard error

23
Finding the 95 Confidence Interval for a
Population Proportion

• A 95 confidence interval for a population
  proportion p is

24
Example Would You Pay Higher Prices to Protect
the Environment?

• In 2000, the GSS asked Are you willing to pay
  much higher prices in order to protect the
  environment?
• Of n 1154 respondents, 518 were willing to do so

25
Example Would You Pay Higher Prices to Protect
the Environment?

• Find and interpret a 95 confidence interval for
  the population proportion of adult Americans
  willing to do so at the time of the survey

26
Example Would You Pay Higher Prices to Protect
the Environment?
27
Sample Size Needed for Large-Sample Confidence
Interval for a Proportion

• For the 95 confidence interval for a proportion
  p to be valid, you should have at least 15
  successes and 15 failures

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95 Confidence

• With probability 0.95, a sample proportion value
  occurs such that the confidence interval contains
  the population proportion, p
• With probability 0.05, the method produces a
  confidence interval that misses p

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How Can We Use Confidence Levels Other than 95?

• In practice, the confidence level 0.95 is the
  most common choice
• But, some applications require greater confidence
• To increase the chance of a correct inference, we
  use a larger confidence level, such as 0.99

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A 99 Confidence Interval for p
31
Different Confidence Levels
32
Different Confidence Levels

• In using confidence intervals, we must compromise
  between the desired margin of error and the
  desired confidence of a correct inference
• As the desired confidence level increases, the
  margin of error gets larger

33
What is the Error Probability for the Confidence
Interval Method?

• The general formula for the confidence interval
  for a population proportion is
• Sample proportion (z-score)(std. error)
• which in symbols is

34
What is the Error Probability for the Confidence
Interval Method?
35
Summary Confidence Interval for a Population
Proportion, p

• A confidence interval for a population proportion
  p is

36
Summary Effects of Confidence Level and Sample
Size on Margin of Error

• The margin of error for a confidence interval
• Increases as the confidence level increases
• Decreases as the sample size increases

37
What Does It Mean to Say that We Have 95
Confidence?

• If we used the 95 confidence interval method to
  estimate many population proportions, then in the
  long run about 95 of those intervals would give
  correct results, containing the population
  proportion

38
A recent survey asked During the last year,
did anyone take something from you by force?

• Of 987 subjects, 17 answered yes
• Find the point estimate of the proportion of the
  population who were victims
• .17
• .017
• .0017

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Section 7.3

• How Can We Construct a Confidence Interval To
  Estimate a Population Mean?

40
How to Construct a Confidence Interval for a
Population Mean

• Point estimate margin of error
• The sample mean is the point estimate of the
  population mean
• The exact standard error of the sample mean is s/
• In practice, we estimate s by the sample standard
  deviation, s

41
How to Construct a Confidence Interval for a
Population Mean

• For large n
• and also
• For small n from an underlying population that is
  normal
• The confidence interval for the population mean
  is

42
How to Construct a Confidence Interval for a
Population Mean

• In practice, we dont know the population
  standard deviation
• Substituting the sample standard deviation s for
  s to get se s/ introduces extra error
• To account for this increased error, we replace
  the z-score by a slightly larger score, the
  t-score

43
How to Construct a Confidence Interval for a
Population Mean

• In practice, we estimate the standard error of
  the sample mean by se s/
• Then, we multiply se by a t-score from the
  t-distribution to get the margin of error for a
  confidence interval for the population mean

44
Properties of the t-distribution

• The t-distribution is bell shaped and symmetric
  about 0
• The probabilities depend on the degrees of
  freedom, df
• The t-distribution has thicker tails and is more
  spread out than the standard normal distribution

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t-Distribution
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Summary 95 Confidence Interval for a
Population Mean

• A 95 confidence interval for the population mean
  µ is
• To use this method, you need
• Data obtained by randomization
• An approximately normal population distribution

47
Example eBay Auctions of Palm Handheld Computers

• Do you tend to get a higher, or a lower, price if
  you give bidders the buy-it-now option?

48
Example eBay Auctions of Palm Handheld Computers

• Consider some data from sales of the Palm M515
  PDA (personal digital assistant)
• During the first week of May 2003, 25 of these
  handheld computers were auctioned off, 7 of which
  had the buy-it-now option

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Example eBay Auctions of Palm Handheld Computers

• Buy-it-now option
• 235 225 225 240 250 250 210
• Bidding only
• 250 249 255 200 199 240 228 255
  232 246 210 178 246 240 245 225
  246 225

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Example eBay Auctions of Palm Handheld Computers

• Summary of selling prices for the two types of
  auctions
• buy_now N Mean StDev Minimum Q1
  Median Q3
• no 18 231.61 21.94 178.00
  221.25 240.00 246.75 yes 7
  233.57 14.64 210.00 225.00 235.00
  250.00
• buy_now Maximum
• no 255.00
• yes 250.00

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Example eBay Auctions of Palm Handheld Computers
52
Example eBay Auctions of Palm Handheld Computers

• To construct a confidence interval using the
  t-distribution, we must assume a random sample
  from an approximately normal population of
  selling prices

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Example eBay Auctions of Palm Handheld Computers

• Let µ denote the population mean for the
  buy-it-now option
• The estimate of µ is the sample mean
• x 233.57
• The sample standard deviation is
• s 14.64

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Example eBay Auctions of Palm Handheld Computers

• The 95 confidence interval for the buy-it-now
  option is
• which is 233.57 13.54 or (220.03, 247.11)

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Example eBay Auctions of Palm Handheld Computers

• The 95 confidence interval for the mean sales
  price for the bidding only option is
• (220.70, 242.52)

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Example eBay Auctions of Palm Handheld Computers

• Notice that the two intervals overlap a great
  deal
• Buy-it-now (220.03, 247.11)
• Bidding only (220.70, 242.52)
• There is not enough information for us to
  conclude that one probability distribution
  clearly has a higher mean than the other

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How Do We Find a t- Confidence Interval for Other
Confidence Levels?

• The 95 confidence interval uses t.025 since 95
  of the probability falls between - t.025 and
  t.025
• For 99 confidence, the error probability is 0.01
  with 0.005 in each tail and the appropriate
  t-score is t.005

58
If the Population is Not Normal, is the Method
Robust?

• A basic assumption of the confidence interval
  using the t-distribution is that the population
  distribution is normal
• Many variables have distributions that are far
  from normal

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If the Population is Not Normal, is the Method
Robust?

• How problematic is it if we use the t- confidence
  interval even if the population distribution is
  not normal?

60
If the Population is Not Normal, is the Method
Robust?

• For large random samples, its not problematic
• The Central Limit Theorem applies for large n,
  the sampling distribution is bell-shaped even
  when the population is not

61
If the Population is Not Normal, is the Method
Robust?

• What about a confidence interval using the
  t-distribution when n is small?
• Even if the population distribution is not
  normal, confidence intervals using t-scores
  usually work quite well
• We say the t-distribution is a robust method in
  terms of the normality assumption

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Cases Where the t- Confidence Interval Does Not
Work

• With binary data
• With data that contain extreme outliers

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The Standard Normal Distribution is the
t-Distribution with df 8
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The 2002 GSS asked What do you think is the
ideal number of children in a family?

• The 497 females who responded had a median of 2,
  mean of 3.02, and standard deviation of 1.81.
  What is the point estimate of the population
  mean?
• 497
• 2
• 3.02
• 1.81

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Section 7.4

• How Do We Choose the Sample Size for a Study?

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How are the Sample Sizes Determined in Polls?

• It depends on how much precision is needed as
  measured by the margin of error
• The smaller the margin of error, the larger the
  sample size must be

67
Choosing the Sample Size for Estimating a
Population Proportion?

• First, we must decide on the desired margin of
  error
• Second, we must choose the confidence level for
  achieving that margin of error
• In practice, 95 confidence intervals are most
  common

68
Example What Sample Size Do You Need For An
Exit Poll?

• A television network plans to predict the outcome
  of an election between two candidates Levin and
  Sanchez
• They will do this with an exit poll that randomly
  samples votes on election day

69
Example What Sample Size Do You Need For An
Exit Poll?

• The final poll a week before election day
  estimated Levin to be well ahead, 58 to 42
• So the outcome is not expected to be close
• The researchers decide to use a sample size for
  which the margin of error is 0.04

70
Example What Sample Size Do You Need For An
Exit Poll?

• What is the sample size for which a 95
  confidence interval for the population proportion
  has margin of error equal to 0.04?

71
Example What Sample Size Do You Need For An
Exit Poll?

• The 95 confidence interval for a population
  proportion p is
• If the sample size is such that 1.96(se) 0.04,
  then the margin of error will be 0.04

72
Example What Sample Size Do You Need For An
Exit Poll?

• Find the value of the sample size n for which
  0.04 1.96(se)

73
Example What Sample Size Do You Need For An
Exit Poll?

• A random sample of size n 585 should give a
  margin of error of about 0.04 for a 95
  confidence interval for the population proportion

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How Can We Select a Sample Size Without Guessing
a Value for the Sample Proportion

• In the formula for determining n, setting
• 0.50 gives the largest value for n
  out of all the possible values to substitute for
• Doing this is the safe approach that guarantees
  well have enough data

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Sample Size for Estimating a Population Parameter

• The random sample size n for which a confidence
  interval for a population proportion p has margin
  of error m (such as m 0.04) is

76
Sample Size for Estimating a Population Parameter

• The z-score is based on the confidence level,
  such as z 1.96 for 95 confidence
• You either guess the value youd get for the
  sample proportion based on other information or
  take the safe approach of setting 0.50

77
Sample Size for Estimating a Population Mean

• The random sample size n for which a 95
  confidence interval for a population mean has
  margin of error approximately equal to m is
• To use this formula, you guess the value youll
  get for the sample standard deviation, s

78
Sample Size for Estimating a Population Mean

• In practice, since you dont yet have the data,
  you dont know the value of the sample standard
  deviation, s
• You must substitute an educated guess for s
• You can use the sample standard deviation from a
  similar study

79
Example Finding n to Estimate Mean Education in
South Africa

• A social scientist plans a study of adult South
  Africans to investigate educational attainment in
  the black community
• How large a sample size is needed so that a 95
  confidence interval for the mean number of years
  of education has margin of error equal to 1 year?

80
Example Finding n to Estimate Mean Education in
South Africa

• No prior information about the standard deviation
  of educational attainment is available
• We might guess that the sample education values
  fall within a range of about 18 years

81
Example Finding n to Estimate Mean Education in
South Africa

• If the data distribution is bell-shaped, the
  range from 3 to 3 will contain
  nearly all the distribution
• The distance 3 to 3 equals 6s
• Solving 18 6s for s yields s 3
• So 3 is a crude estimate of s

82
Example Finding n to Estimate Mean Education in
South Africa

• The desired margin of error is m 1 year
• The required sample size is

83
What Factors Affect the Choice of the Sample Size?

• The first is the desired precision, as measured
  by the margin of error, m
• The second is the confidence level

84
What Other Factors Affect the Choice of the
Sample Size?

• A third factor is the variability in the data
• If subjects have little variation (that is, s is
  small), we need fewer data than if they have
  substantial variation
• A fourth factor is financial
• Cost is often a major constraint

85
What if You Have to Use a Small n?

• The t- methods for a mean are valid for any n
• However, you need to be extra cautious to look
  for extreme outliers or great departures from the
  normal population assumption

86
What if You Have to Use a Small n?

• In the case of the confidence interval for a
  population proportion, the method works poorly
  for small samples

87
Constructing a Small-Sample Confidence Interval
for a Proportion

• Suppose a random sample does not have at least 15
  successes and 15 failures
• The confidence interval formula
• Is still valid if we use it after adding 2 to
  the original number of successes and 2 to the
  original number of failures
• This results in adding 4 to the sample size n