Section 7.2

1
Section 7.2

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

2
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

3
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

4
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

5
Finding the 95 Confidence Interval for a
Population Proportion

• In practice, we use an estimated standard error

6
Finding the 95 Confidence Interval for a
Population Proportion

• A 95 confidence interval for a population
  proportion p is
  

7
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

8
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

9
Example Would You Pay Higher Prices to Protect
the Environment?
10
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
  

11
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

12
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

13
A 99 Confidence Interval for p
14
Different Confidence Levels
15
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

16
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

17
What is the Error Probability for the Confidence
Interval Method?
18
Summary Confidence Interval for a Population
Proportion, p

• A confidence interval for a population proportion
  p is

19
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

20
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

21
Section 7.3

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

22
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

23
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

24
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

25
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

26
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

27
t-Distribution
28
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

29
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?

30
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

31
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
  

32
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

33
Example eBay Auctions of Palm Handheld Computers
34
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

35
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

36
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)

37
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)

38
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
  

39
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

40
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

41
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?

42
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

43
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

44
Cases Where the t- Confidence Interval Does Not
Work

• With binary data
• With data that contain extreme outliers

45
The Standard Normal Distribution is the
t-Distribution with df 8
46
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