What Is a Confidence Interval

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Chapter 21

• What Is a Confidence Interval?

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Thought Question 1
Suppose that 40 of the population disagree with
a proposed new law.
(a) If you randomly sample 10 people, will
exactly four (40) disagree with the law? Would
you be surprised if only two (20) of the people
in the sample disagreed with the law? How about
if none of the sample disagreed?
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Thought Question 2
Suppose that 40 of the population disagree with
a proposed new law.
(b) Now suppose you randomly sample 1000 people,
will exactly 400 (40) disagree with the law?
Would you be surprised if only 200 (20) of the
people in the sample disagreed with the law? How
about if none of the sample disagreed?
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Thought Question 3
A 95 confidence interval for the proportion of
British couples in which the wife is taller than
the husband extends from .02 to .08, or 2 to 8.
What do you think it means to say that the
interval from .02 to .08 represents a 95
confidence interval for the proportion of couples
in which the wife is taller than the husband?
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Thought Question 4
Do you think a 99 confidence interval for the
proportion described in Question 3 would be wider
or narrower than the 95 interval given? Explain.
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Thought Question 5
In a Yankelovich Partners poll of 1000 adults
(USA Today, 20 April 1998), 45 reported that
they believed in faith healing. Based on this
survey, a 95 confidence interval for the
proportion in the population who believe is about
42 to 48. If this poll had been based on 5000
adults instead, do you think the 95 confidence
interval would be wider or narrower than the
interval given? Explain.
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Recall from previous chapters

• Parameter
• fixed, unknown number that describes the
  population
• Statistic
• known value calculated from a sample
• a statistic is used to estimate a parameter
• Sampling Variability
• different samples from the same population may
  yield different values of the sample statistic
• estimates from samples will be closer to the true
  values in the population if the samples are larger

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Recall from previous chapters

• Sampling Distribution
• tells what values a statistic takes and how often
  it takes those values in repeated sampling.

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Case Study
Comparing Fingerprint Patterns
Science News, Jan. 27, 1995, p. 451.
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Case Study Fingerprints

• Fingerprints are a sexually dimorphic
  traitwhich means they are among traits that may
  be influenced by prenatal hormones.
• It is known
• Most people have more ridges in the fingerprints
  of the right hand. (People with more ridges in
  the left hand have leftward asymmetry.)
• Women are more likely than men to have leftward
  asymmetry.
• Compare fingerprint patterns of heterosexual and
  homosexual men.

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Case Study FingerprintsStudy Results

• 66 homosexual men were studied.
• 20 (30) of the homosexual men showed left
  asymmetry.
• 186 heterosexual men were also studied
• 26 (14) of the heterosexual men showed left
  asymmetry.

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Case Study FingerprintsA Question
Assume that the proportion of all men who have
the asymmetry is 15. Is it unusual to observe a
sample of 66 men with a sample proportion ( ) of
30 if the true population proportion (p) is 15?
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Twenty Simulated Samples (n66)
Observed Proportion
Sample Size
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The Rule for Sample Proportions
If numerous samples or repetitions of size n are
taken, the frequency curve of the sample
proportions from various samples will be
approximately bell-shaped. The mean of those
sample proportions will be p (the population
proportion). The standard deviation will be
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Rule Conditions and Illustration

• For rule to be valid, must have
• Random sample
• Large sample size

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The Rule for Sample ProportionsApplied to the
Case Study
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Where should 95 of the sample proportions lie?

• mean plus or minus two standard deviations
• 0.15 ? 2(0.044) 0.062
• 0.15 2(0.044) 0.238
• 95 should fall between 0.062 0.238

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1000 Simulated Samples (n66)
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1000 Simulated Samples (n66)
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1000 Simulated Samples (n30)
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1000 Simulated Samples (n30)
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Confidence Interval for a Population Proportion

• An interval of values, computed from sample data,
  that is almost sure to cover the true population
  proportion.
• We are highly confident that the true
  population proportion is contained in the
  calculated interval.

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Formula for a 95 Confidence Interval for the
Population Proportion (Empirical Rule)

• sample proportion plus or minus two standard
  deviations ofthe sample proportion

• since we dont know the population proportion p
  (needed to calculate the standard deviation) we
  will use the sample proportion in its place.

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Formula for a 95 Confidence Interval for the
Population Proportion (Empirical Rule)
standard error (estimated standard deviation of
)
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Margin of Error
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Formula for a C-level () Confidence Interval for
the Population Proportionwhere z is the
critical value of the standard normal
distribution for confidence level C
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Table 21.1 Common Values of z
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Case Study
Parental Discipline
Brown, C. S., (1994) To spank or not to spank.
USA Weekend, April 22-24, pp. 4-7.
What are parents attitudes and practices on
discipline?
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Case Study Survey
Parental Discipline

• Nationwide random telephone survey of 1,250
  adults.
• 474 respondents had children under 18 living at
  home
• results on behavior based on the smaller sample
• reported margin of error
• 3 for the full sample
• 5 for the smaller sample

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Case Study Results
Parental Discipline

• The 1994 survey marks the first time a majority
  of parents reported not having physically
  disciplined their children in the previous year.
  Figures over the past six years show a steady
  decline in physical punishment, from a peak of 64
  percent in 1988
• The 1994 proportion who did not spank or hit was
  51 !

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Case Study Results
Parental Discipline

• Disciplining methods over the past year
• denied privileges 79
• confined child to his/her room 59
• spanked or hit 49
• insulted or swore at child 45
• Margin of error 5
• Which of the above appear to be different from
  50?

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Case Study Confidence Intervals
Parental Discipline

• denied privileges 79
• p-hat 0.79
• standard error of p-hat
• 95 C.I. .79 2(.019) (.752, .828)
• confined child to his/her room 59
• p-hat 0.59
• standard error of p-hat
• 95 C.I. .59 2(.023) (.544, .636)

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Case Study Confidence Intervals
Parental Discipline

• spanked or hit 49
• p-hat 0.49
• standard error of p-hat
• 95 C.I. .49 2(.023) (.444, .536)
• insulted or swore at child 45
• p-hat 0.45
• standard error of p-hat
• 95 C.I. .45 2(.023) (.404, .496)

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Case Study Results
Parental Discipline

• Asked of the full sample (n1,250) How often
  do you think repeated yelling or swearing at a
  child leads to long-term emotional problems?
• very often or often 74
• sometimes 17
• hardly ever or never 7
• no response 2
• Margin of error 3

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Case Study Confidence Intervals
Parental Discipline

• hardly ever or never 7
• p-hat 0.07
• standard error of p-hat
• 95 C.I. .07 2(.007) (.056, .084)
• Few people believe such behavior is harmless, but
  almost half (45) of parents engaged in it!

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Key Concepts

• Different samples (of the same size) will
  generally give different results.
• We can specify what these results look like in
  the aggregate.
• Rule for Sample Proportions
• Compute and interpret Confidence Intervals for
  population proportions based on sample proportions