suppose we wanted to estimate the proportion of registered voters who are more enthusiastic about...
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Confidence Intervals with Proportions
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Sea Fan
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Suppose we wanted to estimate the proportion of registered voters who are more enthusiastic about voting in this election compared to other years?
Suppose we wanted to estimate the proportion of Dr. Pepper cans that are under-filled?
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Use a single statistic based on sample data to estimate a population parameter
Simplest approachBut not always very precise due to variation in the sampling distribution
Point Estimate
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Are used to estimate the unknown population parameter
Formula:
statistic + margin of error
Confidence intervals
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statistic theof
deviation standard
value
criticalm
Margin of errorShows how accurate we believe our
estimate isThe smaller the margin of error, the
more precise our estimate of the true parameter
Formula:
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Guess my age within 10 years? within 5 years? within 1 year?
Shooting a basketball at a wading pool, will make basket?
Shooting the ball at a large trash can, will make basket?
Shooting the ball at a carnival, will make basket?
Rate your confidence0 - 100
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What happens to your confidence as the interval gets smaller?
The lower your confidence, the smaller the interval. %
%%
%
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Is the success rate of the method used to construct the interval
Using this method, ____% of the time the intervals constructed will contain the true population parameter
Confidence level
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Found from the confidence levelThe upper z-score with probability p
lying to its right under the standard normal curve
Confidence level tail area z*.05 1.645.025 1.96.005 2.576
Critical value (z*)
.05
z*=1.645
.025
z*=1.96
.005
z*=2.57690%95%99%
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Confidence interval for a population proportion:
npp 1p̂ *z
npp ˆ1ˆ
Statistic + Critical value × Standard deviation of the statistic
Margin of error
But do we know the population proportion?
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1. Assumptions
2. Calculations
3. Conclusion
What are the steps for performing a confidence interval?
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SRS of contextApproximate Normal distribution becausenp > 10 & n(1-p) > 10
Population is at least 10n
Conditions:Where are the last two assumptions from?
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We are ________% confident that the true proportion context is between ______ and ______.
Statement: (memorize!!)
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A May 2000 Gallup Poll found that 38% of a random sample of 1012 adults said that they believe in ghosts. Find a 95% confidence interval for the true proportion of adults who believe in ghost.
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Conditions:
•Have an SRS of adults
•np =1012(.38) = 384.56 & n(1-p) = 1012(.62) = 627.44 Since both are greater than 10, the distribution can be approximated by a normal curve
•Population of adults is at least 10,120.
41,.35.1012
)62(.38.96.138.
1*ˆ
npp
zP
We are 95% confident that the true proportion of adults who believe in ghosts is between 35% and 41%.
Step 1: check conditions!
Step 2: make calculations
Step 3: conclusion in context
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The manager of the dairy section of a large supermarket took a random sample of 250 egg cartons and found that 40 cartons had at least one broken egg. Find a 90% confidence interval for
the true proportion of egg cartons with at least one broken egg.
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Conditions:
•Have an SRS of egg cartons
•np =250(.16) = 40 & n(1-p) = 250(.84) = 210 Since both are greater than 10, the distribution can be approximated by a normal curve
•Population of cartons is at least 2500.
198,.122.250
)84(.16.645.116.
We are 90% confident that the true proportion of egg cartons with at least one broken egg is between 12.2% and 19.8%.
Step 1: check conditions!
Step 2: make calculations
Step 3: conclusion in context