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