๐: ESTIMATES, CONFIDENCE INTERVALS, AND...
Transcript of ๐: ESTIMATES, CONFIDENCE INTERVALS, AND...
The CLT for ๐Estimating proportion
Hypothesis on the proportion
Old exam question
Further study
CONTENTS
โช Estimating, confidence intervals, and hypothesis test for ๐are based on the central limit theoremโช and therefore on the normal distribution
โช For ๐2 we needed another distributionโช the ๐2-distribution
โช What to use for ๐?โช the probability of success in a Bernoulli experiment
โช Based on sampling theoryโช so, repeated Bernoulli experiment
โช so, a binomial distribution
โช and for large ๐, approximately a normal distribution (โ CLT)
THE CLT FOR ๐
Define ๐๐ as the outcome (0 or 1) in one Bernoulli experiment
โช Total number of โ1โs in ๐ Bernoulli experimentsโช ๐ = ฯ๐=1
๐ ๐๐โช Average number of โ1โs (due to CLT, with binomial results):
โช ๐ =๐
๐= เดค๐~๐ ๐๐,
๐๐2
๐= ๐ ๐,
๐ 1โ๐
๐
โช provided ๐๐ โฅ 5 and ๐ 1 โ ๐ โฅ 5
THE CLT FOR ๐
๐ is the estimator of ๐a concrete estimate is ๐
Estimator:
โช for ๐: เดค๐~๐ ๐๐,๐๐2
๐
โช for ๐: ๐~๐ ๐,๐ 1โ๐
๐
Point estimate:
โช for ๐: เท๐ = าง๐ฅ =1
๐ฯ๐=1๐ ๐ฅ๐, with observation ๐ฅ๐ โ โ
โช for ๐: เท๐ = ๐ =1
๐ฯ๐=1๐ ๐ฅ๐, with observation ๐ฅ๐ = 0 or 1
Standard error of estimate:
โช for ๐: ๐ เดค๐ =๐๐
๐
โช for ๐: ๐๐ =๐ 1โ๐
๐
THE CLT FOR ๐
Both standard errors decrease with ๐
โช Estimating ๐ by ๐
โช and estimating ๐๐ =๐ 1โ๐
๐by ๐ ๐ =
๐ 1โ๐
๐
โช standard error of proportion
โช So, we have for ๐โช a point estimate ๐ =
๐
๐
โช an interval estimate ๐ โ ๐ง๐ผ/2๐ 1โ๐
๐, ๐ + ๐ง๐ผ/2
๐ 1โ๐
๐
โช 1 โ ๐ผ confidence interval for ๐
โช ๐ โ ๐ง๐ผ/2๐ 1โ๐
๐โค ๐ โค ๐ + ๐ง๐ผ/2
๐ 1โ๐
๐
ESTIMATING PROPORTION
Example
Context: a sample of 75 retail in-store purchases showed that 24were paid in cash. Give a 95% confidence interval for ๐.
โช ๐ =๐ฆ
๐=
24
75= 0.32; this is the point estimate for ๐
โช standard error of the estimate:
โช ๐ ๐ =๐ 1โ๐
๐=
0.32 1โ0.32
75= 0.054
โช CI๐,0.95: โช 0.32 โ 1.96 ร 0.054 , 0.32 + 1.96 ร 0.054 = 0.214 , 0.426โช or: 0.214 โค ๐ โค 0.426โช or: 0.32 ยฑ 0.106
ESTIMATING PROPORTION
Check validity: ๐๐ โฅ 5 and ๐ 1 โ ๐ โฅ 5
You flip a coin 100 times and find 45 times head. Give a
95% confidence interval for ๐โ๐๐๐ .
EXERCISE 1
Test a hypothesis on the proportion of a Bernoulli process
โช Example:โช you are a police officer
โช you wonder if less than 50% of the (one-sided) traffic accidents
occur with female drivers driving the car
HYPOTHESES ON THE PROPORTION
โช Statistical modelโช each accident has an underlying Bernouilli process of happening
to a man (0) or to a woman (1), ๐~๐๐๐ก ๐โช you observe the next ๐ = 5 car accidents, and report the
outcomes (0/1)
โช you define ๐ as the number of accidents that is caused by a
woman
โช the sequence of 5 observations can be regarded as a binomial
process, ๐~๐ต๐๐ ๐, 5โช you start by assuming the accident rates are equal, i.e.,
hypothesize that ๐ = 0.5
โช Suppose you observed ๐ฆ = 1, i.e., one car accident by a
woman
HYPOTHESES ON THE PROPORTION
โช Step 1:โช ๐ป0: ๐ โฅ 0.5; ๐ป1: ๐ < 0.5; ๐ผ = 0.05
โช Step 2:โช sample statistic: ๐ =#female; reject for โtoo smallโ values
โช Step 3:โช if ๐ป0 is just true, ๐~๐ต๐๐ 0.5,5 ; no assumptions required
โช Step 4:โช ๐โvalue = ๐๐ต๐๐ 0.5,5 ๐ โค 1 = ๐ ๐ = 0 + ๐ ๐ = 1 =
0.0313 + 0.1563 = 0.1876
โช Step 5:โช ๐โvalue > ๐ผ ; do not reject ๐ป0; there is not sufficient evidence
for concluding that ๐ < 0.5
HYPOTHESES ON THE PROPORTION
What if we have a large sample, say ๐ = 100?
โช binomial tables and formulas donโt work
Use normal approximation
โช if ๐~๐ต๐๐ ๐, ๐ then ๐ =๐โ๐๐
๐๐ 1โ๐~๐ 0,1
โช conditions: ๐๐ โฅ 5 and ๐ 1 โ ๐ โฅ 5: OK
Example
โช same as before (car accidents by gender)
โช but now based on ๐ = 100โช with ๐ฆ = 40 observed accidents by women
HYPOTHESES ON THE PROPORTION
โช Step 1:โช ๐ป0: ๐ โฅ 0.5; ๐ป1: ๐ < 0.5; ๐ผ = 0.05
โช Step 2:โช sample statistic: ๐ =#female; reject for โtoo smallโ values
โช Step 3:
โช if ๐ป0 is just true, ๐ =๐โ๐๐
๐๐=
๐โ๐๐
๐๐ 1โ๐~๐ 0,1
โช normal approximation OK (๐๐ โฅ 5 and ๐ 1 โ ๐ โฅ 5)
โช Step 4:
โช ๐ง๐๐๐๐ =40โ100ร0.5
100ร0.5 1โ0.5= โ2.00 (see, however, next page!)
โช ๐ง๐๐๐๐ก = โ1.645
โช Step 5:โช reject ๐ป0, accept ๐ป1; there is sufficient evidence for concluding that ๐ < 0.5
HYPOTHESES ON THE PROPORTION
โช Note:โช we forgot about the continuity correction
โช a slightly more accurate result can be achieved with the continuity
correction
โช Example:
โช ๐ ๐ โค 40 โ ๐ ๐ โค 401
2= ๐ ๐ โค
401
2โ100ร0.5
100ร0.5ร 1โ0.5=
๐ ๐ โค โ1.9 < 0.05
โช When needed?โช not when ๐โvalue = 0.002 or ๐โvalue = 0.743โช but required in cases like the example, when ๐โvalue โ ๐ผ
HYPOTHESES ON THE PROPORTION
Doane & Seward 5/E 11.1-11.2
Tutorial exercises week 5
confidence intervals
hypothesis tests (binomial)
hypothesis tests (normal)
FURTHER STUDY