Estimating Proportions with Confidence

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Chapter 19

• Estimating Proportions with Confidence

2
Thought Questions
1, page 335
One example we will see in this chapter is a 95
confidence interval for the proportion of British
couples in which the wife is taller than the
husband. The interval 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 Questions
2, page 335
Do you think a 99 confidence interval for the
proportion described in Question 1 would be wider
or narrower than the 95 interval given? Explain.
4
Thought Questions
3, page 335
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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Thought Questions
4, page 335
How do you think the concept of margin of error,
explained in Chapter 4, relates to confidence
intervals for proportions? As a concrete
example, can you determine the margin of error
for the situation in Question 1 from the
information given? In Question 3?
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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.

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

• sample proportion plus or minus two standard
  deviations of the 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)
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Margin of Error
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Formula for a 100?(1-a) Confidence Interval for
the Population Proportion where z?/2 is the
standardized score with ?/2 probability above it
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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

• Compute confidence intervals for population
  proportions based on one sample.
• Interpret confidence intervals for proportions