Embed Size (px)
Citation preview
7-1Esti
m
Unit 7
Statistical Inference - 1Estimation
FPP Chapters 21,23, 26-29 Point Estimation
Margin of Error
Interval Estimation
- Confidence Intervals
Sample Size Computations
Next:
Statistical Tests of Hypotheses
-Hypothesis Testing A.05
7-2Esti
m
EstimationBox models:
If we know what goes in the box, then we can say how likely various outcomes are.
In practice,we do not know what is in the box.
We do not know the population parameters.Insteadwe use data to estimate the population parameters, such as average, %, sd, …
That is, we infer the population parameters, based on the sample of data.
We make INFERENCE from the SAMPLE (data) to the POPULATION.
7-3Esti
m
A Model for Estimation
Sample Value = Parameter Value + (Bias) +
Chance Error
Thus,Estimate =
Parameter Value + (Bias) + Chance Error
Size of Chance Errors depends partly upon the sampling procedure-----------------------
Recall:
Population Sample Parameter Statistic
For now, assume random sampling, 100% response rate, and correct responses.
7-4Esti
m
Margin of ErrorPoint estimate:
To estimate the population average (mean) with a single value, use
The likely size of your estimation error is
Margin of Error = some multiple of SE
Ex 1: Margin of error for estimating the population average by the sample average is proportional to SE(avg).
Ex 2: Margin of error for estimating the population percent by the sample percent is proportional to SE(percent).
7-5Esti
m
Newspaper Survey ExampleAbout the poll
This poll was conducted for The Seattle Times by Elway Research of Seattle. Pollsters contacted 403 randomly selected adults across the state by telephone April 6-11.The geographic distribution of the respondents was reflective of the population statewide. The poll has a margin of error of 5 percent, meaning that, in theory, results have a 95 percent chance of coming within 5 percentage points of results that would have been obtained had all adults in the state been interviewed.
7-6Esti
m
7-7Esti
m
Interval EstimationCombining Point Estimation & Margin
of Error
Interval estimate:
Rather than give a single estimated value for the parameter, give instead an estimated interval of values.
This combinespoint estimationmargin of error
Approximate level 68% confidence interval:
sample estimate +/- 1 SE(estimate)
Approximate level 95% confidence interval:
sample estimate +/- 2 SE(estimate)
Approximate level ____% confidence interval:
sample estimate +/- 2 ___ SE(estimate)
7-8Esti
m
Confidence IntervalsA confidence interval is used
when estimating an unknown parameter from sample data. The interval gives a range for the parameter - and a confidence level that the range covers the true value.
The width of the interval depends upon how confident you want to be that your interval includes the population parameter value.
Chances are in the sampling procedure, not in the parameter.
7-9Esti
m
Confidence IntervalExample-1
Course credits
7-10Esti
m
Point estimate:
Our group’s point estimate of the population average is:
The likely size of our estimation error is:
Interval estimate:
Our approximate level 68% confidence interval for the population average is:
Confidence Intervals- Ex 1
7-11Esti
m
Confidence Interval
Example- 2Upper division
7-12Esti
m
Point estimate:
Our group’s point estimate of the population proportion is:
The likely size of our estimation error is:
Interval estimate:
Our approximate level 68% confidence interval for the population proportion is:
Confidence Intervals- Ex 2
7-13Esti
m
The BootstrapWhen estimating a
population percentage (i.e. when sampling from a 0-1 box), the fraction of 0’s and 1’s in the box is unknown.
The SD of the box can be estimated by substituting the fraction of 0’s and 1’s in the sample for the unknown fractions in the box.
The estimate is good when the sample is reasonably *large*.
7-14Esti
m
Confidence Interval
Example- 340,000 students enrolled
Simple random sample of 900, of whom 630 “grew up in Washington”
Want to estimate % UW students who “grew up in WA”
Use as estimate:
7-15Esti
m
The Box Model
Confidence Intervals- Ex 3
7-16Esti
m
If we draw another sample of 900 students, what is the chance that the observed sample % is between 65% and 75%?
Confidence Intervals- Ex 3
7-17Esti
m
The Bootstrap again
We don’t know what fraction of the population “grew up in WA”.
So we estimate it& substitute our estimate into the formulae in place of the actual “true” fraction.
7-18Esti
m
Bootstrapping
.
7-19Esti
m
An approximate level ____ % confidence level for the percent of UW students who “grew up in Washington” is …
Confidence Intervals- Ex 3
7-20Esti
m
Assumptions for a
Confidence Interval
Before using this procedure for constructing an approx level ____ % CI, check that the following conditions are met.
• simple random sample,• either the population histogram is approximately normal or the sample size is sufficiently large for the Central Limit Theorem to give us approximate normality
7-21Esti
m
Interpreting a Confidence Interval
IF the appropriate conditions are met, & we construct an approximate ____% confidence interval.
We can be about ___% confident that this interval contains the true population parameter value.
7-22Esti
m
Interpreting a Confidence Interval
The CI depends on the sample.The confidence level depends upon the procedure used (multiplier for the SE).
For about 95% of all samples, the interval
sample ____ +/- 2 SE(____)
covers the population _____, and for the other 5% it fails.
The chances are in the sampling procedure, not in the parameter.The parameter is a fixed number.
7-23Esti
m
No !! NO !! NO !!
NO !! NO !! NO !! NO !! NO !! NO !!
“There is a 95% chance that the parameter (population %) falls inside my interval, 46% +/ 3%.”
NO !! NO !! NO !! NO !! NO !! NO !!
7-24Esti
m
Warning !
Check carefully that the appropriate conditions are met BEFORE applying any statistical procedure - including the construciton of Confidence Intervals.
These confidence inteval methods are for simple random samples, and should NOT be used for other kinds of samples !
7-25Esti
m
Interpreting a Confidence Interval
One of these is WRONG and one of these is CORRECT.
UW enrollment - % “Washingtonians” ex.
Approx level 95% Confidence interval
[67%, 73%]
“There is a 95% chance that the population % is between 67% and 73%.”
“For about 95% of all samples, the interval
sample % +/- 2 SE(%)Covers the population percentage.For the other 5% it fails.”
7-26Esti
m
Confidence Levels
are ApproximateBecause(1) Using the Normal
Approximation(2)SE is estimated (unknown)
To use these procedures, check- good simple reandom sampel- If the percentage is near 0%
or 100%, then need much larger sample than if the percentage is near 50%.
CHECK THESE CONDITIONS !!!
7-27Esti
m
Sample Size
Computations
7-28Esti
m
Another Confidence Interval ExampleA manufacturing process for
bricks is known to give an output whose weights have sd 0.12 pounds, regardless of the mean weight. A random sample of 100 bricks is selected from today’s output. The sample mean is 4.07 pounds.
Construct an approximate level 95% confidence interval for the mean weight of today’s brick production.
Newb.88.295
7-29Esti
m
And Another Confidence
Interval ExampleA personnel manager knows that
historically, the scores on aptitude tests given to candidates for trainee positions have vollowed a normal distribution with SD 28.2.
A simple random sample of 30 test scores for the curent year’s applicants was taken and found to have a sample mean of 122.0.
Construct an approximate level 95% confidence interval for the mean score for all of this year’s applicants.
Newb.8 295
7-30Esti
m
Bootstrap Example
The cloze readability procedure is designed to measure the effectiveness of a written communication. Research has indicated that a score of 0.57 or more on the cloze test demonstrates adequate understandability of the written material. A random sample of 352 certified public accountants was asked to read financial report messages. The sample mean score was 0,6041 and the sample SD was 0.1128.
Construct an approximate level 95% confidence interval for the population mean score.
