Download 8.1 Confidence Intervals Notes

8.1 Confidence Intervals: The Basics
Estimation: The assignment of values to a population parameter based on a value of the corresponding sample statistic.
1. Estimator or Point Estimator: A sample statistic used to estimate a population parameter.
Parameter: ?? We don't know what this is
a) Maybe it's too expensive to find, or impossible to collect.
b) So we take a sample and find a statistic!
Ex: If μ = population mean
then the Estimator is x(bar) ­ the sample mean
If we collect data and find x(bar) = 5
then x(bar) = 5 becomes an Estimate or Point Estimate.
Steps for Estimation
1. Select a Sample
2. Collect the required measurement from the sample (height, weight, etc. )
3. Calculate the value of the sample statistic
4. Assign value or values to the corresponding population parameter.
These values are Unbiased but not consistent
xxx
xxx
xxx
xxx
μ
These values are unbiased (on­
target) and consistent (together)
­they are estimating what we want to estimate!
These values are Consistent but Biased
(off target)
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Definition: Point Estimator and Point Estimate
A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic from a sample is called a point estimate. Ideally, a point estimate is our "best guess" at the value of an unknown parameter.
Example: x(bar) = 5
s (sample standard deviation)= 1.4
p­hat (sample proportion) = .75
Take a statistic and set it equal to one value. That is a point estimate!
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Example: P470 From Batteries to Smoking
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THE IDEA OF A CONFIDENCE INTERVAL
Example: Mystery Mean p471
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Definition: Confidence Interval, Margin of Error, Confidence Level
A confidence interval for a parameter has two parts:
1. An interval calculated from the data, which has the form
estimate ± margin of error
The margin of error tells how close the estimate tends to be to the unknown parameter in repeated random sampling.
2. A confidence level C, which gives the overall success rate of the method for calculating the confidence interval. That is, in C% of all possible samples, the method would yield an interval that captures the true parameter value. There are several ways to write a confindence interval:
240.79 ±10 or 230.79≤μ≤250.79 or (230.79, 250.79)
Other Definitions:
Interval Estimate: An interval constructed around the point estimate. It is stated that this interval is likely to contain the population parameter.
Margin of Error ­ The # added to and subtracted from the point estimate.
Point Estimate ± Margin of Error
Confidence Interval How much confidence we have that this interval contains the true population parameter.
We usually choose a confindence interval of above 90% b/c we want to sure of our conclusions. 95% is the most common confindence interval.
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Example: 95% Confidence Interval
There is a Probability of .95 that a statistic will differ from the parameter by less than the margin of error.
a) Wrong Interpretation: The parameter has a 95% chance of falling inside the confidence interval.
b) Right Interpretation: The methods used to produce the confidence intervals, in the long run, will contain the parameter in 95% of the random samples collected.
c) We Say: "With 95% confidence, the parameter falls inside the lower and upper confidence limits."
This is the most common confidence interval!
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Activity: p473 The Confidence Interval Applet
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Interpreting Confidence Levels and Confidence Intervals
Confidence Levels: To say that we are 95% confident is shorthand for "95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter.
Confidence Interval: to interpret a C% confidence interval for an unknown parameter say, "We are C% confident that the interval from _____ to _____captures the actual value of the [population parameter in context].
Plausible does not mean the same thing as possible!! Some would argue that just about any value of a parameter is possible. A plausible value of a parameter is a reasonable or believable value based on the data.
Caution: The confidence level does not tell us the chance that a particular confidence interval captures the population parameter. Instead, the confidence interval gives us a set of plausible values for the parameter.
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Example: the Mystery Mean p475
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Think About It! p475
What's the probability that our 95% confidence interval captures the parameter?
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Example: Do You Use Twitter? p476
AP Exam Tip: On a given problem, you may be asked to interpret the confidence interval, the confidence level, or both. Be sure you understand the difference: the confidence level describes the long­run capture rate of the method, and the confidence interval gives a set of plausible values for the parameter.
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Check Your Understanding p476
Activity: The confidence interval applet p477
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CONSTRUCTING A CONFIDENCE INTERVAL
Book Defn:
Calculating a Confidence Interval
The confidence interval for estimating a population parameter has the form
margin of error
statistic ± (critical value) x (standard deviation of statistic)
where the statistic we use is the point estimator for the parameter.
margin of error
Black Book Defn:
LOGIC OF A CONFIDENCE INTERVAL
1. Point Estimate ±Confidence Level x Standard Error (standard deviation of statistic)
Confidence Interval for μ standard error
x(bar) ± (Confidence) x (σ/√n)
margin of error
1. As a Confidence Level Increases, the margin of error increases
result: A wider Confidence Interval
2. As Sample size Increases, the Margin of Error will decrease
result: A smaller Confidence Interval
b/c √n is increasing
Read p478
Recall that σx = σ/√n and σphat = √p(1­p)/n So as the sample size n increases, the standard deviation of the statistic decreases.
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USING CONFIDENCE INTERVALS WISELY
3 IMPORTANT CONDITIONS FOR CONSTRUCTING CONFIDENCE INTERVALS
1. Random: The data come from a well­designed random sample or randomized experiment.
2. Normal: The sampling distribution of the statistic is approximately Normal.
3. Independent: Individual observations are independent. Wehn sampling without replacement, the sample size n should be no more than 10% of the population size N (the 10% condition) to use the formula for the standard deviation of the statistic.
Two Important Reminders:
1. Our method of calculation assumes that the data come from an SRS of size n from the population of interest.
2. The margin of error in a confidence interval covers only chance variation due to random sampling or random assignment.
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Homework:
p481
5­13odd
p481 and 496
17, 19­24, 27, 31, 33
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