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## Ch10.pps

**Principal Idea: **
**Estimating **
**Survey 150 randomly selected students and **

41% think marijuana should be legalized.
**Proportions **
If we report

**between 33% and 49% of all students** at

**Chapter 10 **
the college think that marijuana should be legalized,

**how confident can we be that we are correct**?

**Confidence **
**Confidence interval: **an interval of estimates

that is likely to capture the population value.

**Objective**: how to calculate and interpret a confidence

interval estimate of a population proportion.

Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**10.1 The Language and **
**More Language and Notation of Estimation **
** Notation of Estimation **
•

**Population proportion: **the fraction of the population

that has a certain trait/characteristic or the probability
•

**Unit: **an individual person or object to be measured.

of success in a binomial experiment – denoted by

*p*.
•

**Population **(or

**universe**): the entire collection of units

The value of the

*parameter p *is not known.
about which we would like information or the entire
collection of measurements we would have if we could

**Sample proportion: **the fraction of the sample

that has a certain trait/characteristic – denoted by .
measure the whole population.
The

*statistic *is an estimate of

*p. *
•

**Sample: **the collection of units we will actually measure

or the collection of measurements we will actually obtain.
The

**Fundamental Rule for Using Data for Inference **is

•

**Sample size: **the number of units or measurements in the

that available data can be used to make inferences about
sample, denoted by

*n*.
a much larger group

*if the data can be considered to be representative with regard to the question(s) of interest. *
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**10.2 Margin of Error **
**Example 10.1 ***Teens and Interracial Dating *
**Media Descriptions of Margin of Error: **
**1997 ***USA Today***/Gallup Poll of teenagers across country: **
**57% of the 497 teens who go out on dates say they've been **
• The difference between the sample proportion

**out with someone of another race or ethnic group. **
and the population proportion is

**less** than the

margin of error

**about 95% of the time**, or for

Reported

**margin of error** for this estimate was about

**4.5%**.

about 19 of every 20 sample estimates.
• In surveys of this size, the difference between

*the sample *
*estimate *of 57% and

*the true percent *is likely
The difference between the sample proportion
than 4.5% one way or the other.
and the population proportion is

**more** than the

margin of error

**about 5% of the time**, or for

• There is, however, a small chance that the sample
about 1 of every 20 sample estimates
estimate might be off by more than 4.5%.
* The value of how ‘likely' is often 95%.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**Example 10.2 ***If I Won the Lottery … *
**10.3 Confidence Intervals **
If you won 10 million dollars in the lottery,
would you

**continue to work or stop working**?

**Confidence interval: **an interval of values computed from

sample data that is

*likely* to include the true population value.

1997 Gallup Poll: 59% of the 616 employed respondents
said they would continue to work.

**Interpreting the Confidence Level **
Reported information about this poll:
• The

**confidence level **is the probability that the procedure

used to determine the interval

*will* provide an interval that
• Results based on telephone interviews with a randomly
includes the population parameter.
selected sample of 1014 adults, conducted Aug 22–25, ‘97.
• If we consider

*all* *possible* randomly selected samples of
• Among this group, 616 are employed full-time/part-time.
same size from a population, the

*confidence level *is the
fraction or percent of those samples for which the
• For results based on this sample of "workers," one can say
confidence interval includes the population parameter.
with

*95% confidence* that the

**error** attributable to

sampling could be

**plus or minus 4 percentage points**.

**Note: **Often express the confidence level as a percent.

Common levels are 90%, 95%, 98%, and 99%.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**Constructing a 95% Confidence Interval **
**Example 10.1 ***Teens and *
**for a Population Proportion **
** Interracial Dating (cont) **
**Poll**: 57% of dating teens sampled had gone out

**Sample estimate **±

** Margin of error **
with somebody of another race/ethnic group.
Margin of error was

**4.5%**.

In the

*long run*, about

**95% of all confidence intervals**
computed in this way will capture the population value

**95% Confidence Interval: **
of the proportion, and about 5% of them will miss it.
57% ± 4.5%, or 52.5% to 61.5%

**Be careful**: The confidence level only expresses

We have

**95% confidence** that somewhere between

how often the procedure works in the long run.
Any one specific interval either does or does not
52.5% and 61.5% of all American teens who date have
include the true unknown population value.
gone out with somebody of another race or ethnic group.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**Example 10.2 ***Winning the Lottery *
**10.4 Calculating A Margin of **
** and Work (cont) **
** Error for 95% Confidence **
**Poll**: 40% of employed workers sampled would

quit working if they won the lottery.
Margin of error was

**4%**.

For a

**95%** confidence level, the approximate

**95% Confidence Interval Estimate: **
**margin of error** for a sample proportion is

** Sample estimate **±

** Margin of error **
** 36% to 44% **
With 95% confidence, somewhere between 36% and 44% of working Americans would say they would quit working

**Note**: The "95% margin of error" is simply

if they won $10 million in the lottery.
two standard errors, or 2

*s.e.*( ).
Interval does not cover 50% => Appears that fewer than half of all working Americans think they would quit if won lottery.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**Factors that Determine Margin of Error **
**Example 10.3 ***Pollen Count Must Be High *
**1. ***The sample size, n. *
**Poll**: Random sample of 883 American adults.

When sample size increases, margin of error decreases.
"

**Are you allergic to anything?**"

**2. ***The sample proportion, ***. **
**Results**: 36% of the sample said "yes"

If the proportion is close to either 1 or 0 most individuals have the same trait or opinion, so there
Find a 95% confidence interval.
is little natural variability and the margin of error is smaller than if the proportion is near 0.5.

**3. ***The "multiplier" ***2. **
Connected to the "95%" aspect of the margin of error.

* *
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**The Conservative Estimate **
**Example 10.3 ***Really Bad Allergies (cont) *
** of Margin of Error **
**Poll**: Random sample of 883 American adults

**Conservative estimate **
3% of the sample experience "

**severe**" symptoms.

** of the margin of error **=

Find both the 95% confidence interval and the 95%
conservative confidence interval.
• It usually

**overestimates** the actual size of the

margin of error.
• It works (

**conservatively**) for all survey questions

based on the same sample size, even if the sample proportions differ from one question to the next.
• Obtained when = .5 in the margin of error formula.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**10.5 General Theory of CIs **
**General Description of the **
** for a Proportion **
**Approximate 95% CI for a Proportion **
**Developing the 95% Confidence Level **
**Approximate 95% CI for the population proportion**:

From the sampling distribution of we have:
± 2 standard errors
For 95% of all samples,
The standard error is
-2 standard deviations < –

*p *< 2 standard deviations Don't know true standard deviation, so use standard error.

**Interpretation**: For about 95% of all randomly selected

For approximately 95% of all samples,
samples from the population, the confidence interval
-2 standard errors < –

*p *< 2 standard errors
computed in this manner captures the population proportion.
which implies for approximately 95% of all samples,

**Necessary Conditions**: and are both greater

– 2 standard errors <

*p *< + 2 standard errors
than 10, and the sample is randomly selected.

* *
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**General Format for **
**More about the Multiplier **
** Confidence Intervals **
For any confidence level, a confidence interval for either a population proportion or a population mean can be expressed as

**Sample estimate **±

** Multiplier **×

** Standard error **
The multiplier is affected by

**Note**: Increase confidence level => larger multiplier.

the choice of confidence level.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**Formula for a Confidence Interval **
**Example 10.6 ***Intelligent Life Elsewhere? *
** for a Population Proportion ***p*
**Poll**: Random sample of 935 Americans

Do you think there is intelligent life on other planets?

**Results**: 60% of the sample said "yes". Find the

90%, 95%, and 98% confidence interval.
• is the sample proportion. •

*z** denotes the multiplier. where
• is the standard error of .

**Q**: does the 95% interval contain the value p=0.5?

Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**Conditions for Using the Formula **
**10.6 Choosing a Sample Size **
1. Sample is

**randomly selected** from the population.

**Note**: Available data can be used to make inferences

**95% conservative **
about a much larger group

*if the data can be *
**margin of error **
*considered to be representative with regard to *
*the question(s) of interest. *
sample sizes

*n*
2. Normal curve approximation to the distribution
of possible sample proportions assumes a

**Important features: **
**"large" sample size**. Both and

should be at least 10 (although some say these

1. When sample size is

*increased, *margin of error

*decreases. *
need only to be at least 5).
2. When a large sample size is made even larger, the
improvement in accuracy is relatively small.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**The Effect of Population Size **
**10.7 Using Confidence Intervals **
** to Guide Decisions **
For most surveys, the number of people in

the population has almost

**no influence*** on

**Principle 1. **A

**value** **not** **in** a confidence interval can be

the accuracy of sample estimates.

**rejected** as a possible value of the population proportion.

A

**value** **in** a confidence interval is an "

**acceptable**"

**Margin of error for a sample size of 1000**
possibility for the value of a population proportion.
is about

**3%** **whether** the number of people

**Principle 2. **When the confidence intervals for

in the

**population is 30,000 or 200 million**.

proportions in two different populations

**do not overlap**,

it is reasonable to

**conclude** that the

**two population **
* As long as the population is at least

**proportions are different**.

*ten* times as large as the sample.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**Example 10.7 ***Which Drink Tastes Better? *
**Case Study 10.1 ***ESP Works with Movies *
**Taste Test**: A sample of

**60 people** taste both drinks

and

**55% like taste of Drink A** **better** than Drink B.

**ESP Study by Bem and Honorton (1994) **
Makers of Drink A want to advertise these results.
• Subjects (receivers) described what another person
Makers of Drink B make a 95% confidence interval
(sender) was seeing on a screen.
for the population proportion who prefer Drink A.
• Receivers shown 4 pictures, asked to pick which they
thought sender had actually seen.

**Is **Drink A is preferred by the

**majority** of the population

• Actual image shown randomly picked from 4 choices.
(represented by the sample)?
• Image was either a single, "static" image or a "dynamic"
short video clip, played repeatedly (additional three choices shown were always of the same type as actual.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**Case Study 10.1 ***ESP Works (cont) *
**Case Study 10.2 ***Nicotine Patches vs Zyban *
**Bem and Honorton (1994) ESP Study Results **
**Study: **New England Journal of Medicine (3/4/99)

• 893 participants

**randomly allocated** to

**four treatment groups**: placebo, nicotine

patch only, Zyban only, and Zyban plus

Is there

**enough evidence** to say that the

**% of correct guesses**
for dynamic pictures is

**significantly above 25%**?

• Participants

**blinded**:

all used a patch (nicotine or placebo) and all took a pill (Zyban or placebo).
• Treatments used for

**nine** weeks.

Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**Case Study 10.2 ***Nicotine (cont) *
**Case Study 10.3 ***What a Great Personality *
**Would you date someone** with a great personality

even though you did

**not** find them attractive?

**Women**: 61.1% of 131 answered "yes."

**Zyban is effective**
95% confidence interval is 52.7% to 69.4%.
(no overlap of Zyban
42.6% of 61 answered "yes."
and no Zyban CIs)
95% confidence interval is 30.2% to 55%.

**Nicotine patch is not **
**particularly effective**
•

**Higher proportion** of

(overlap of patch

**women** would say yes.

* *
and no patch CIs)
CIs slightly overlap
• Women CI

**narrower**
than men CI due to larger sample size
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

**In Summary****: Confidence Interval **

for a Population Proportion *p*
**General CI for ***p***: **
**Approximate **
**95% CI for ***p***: **
**95% CI for ***p***: **
Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Source: http://www.math.yorku.ca/~hkj/Teaching/1500/Coverage/ch10.pdf

JOURNAL OF RESEARCH ON ADOLESCENCE, 21(1), 180 – 195 Puberty and Its Measurement: A Decade in Review Lorah D. Dorn and Frank M. Biro Cincinnati Children's Hospital Medical Center and University of Cincinnati Since the early 1980s, the focus on the importance of puberty to adolescent development has continued with variability inthe methodology selected to measure puberty. To capture the relevant and important issues regarding the measurement ofpuberty in the last decade, this paper will address (1) the neuroendocrine aspects of puberty and its components, as well asthe timing of puberty and its tempo; (2) why puberty is measured, including the relevance of puberty and its timing tohealth and development as well as the relevance of being off-time, that is, early or late with respect to a reference group; (3)the measurement of puberty and its methodology with respect to pubertal staging by physical examination, self-reportmeasures, and their agreement with other methods and measures, hormones and their methods of measurement, andcomparison of hormone concentrations to pubertal stage; and (4) recommendations for what is needed in the next decaderegarding the measurement of puberty.

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