π: ESTIMATES, CONFIDENCE INTERVALS, AND TESTS Busiess Statistics
CONTENTS The CLT for π Estimatig proportio Hypothesis o the proportio Old exam questio Further study
THE CLT FOR π Estimatig, cofidece itervals, ad hypothesis test for μ are based o the cetral limit theorem ad therefore o the ormal distributio For σ 2 we eeded aother distributio the χ 2 -distributio What to use for π? the probability of success i a Beroulli experimet Based o samplig theory so, repeated Beroulli experimet so, a biomial distributio ad for large, approximately a ormal distributio ( CLT)
THE CLT FOR π Defie X i as the outcome (0 or 1) i oe Beroulli experimet Total umber of 1 s i Beroulli experimets Y = σ i=1 X i Average umber of 1 s (due to CLT, with biomial results): P = Y = തX~N μ X, σ X 2 = N π, provided π 5 ad 1 π 5 π 1 π P is the estimator of π a cocrete estimate is p
THE CLT FOR π Estimator: for μ: തX~N μ X, σ X 2 for π: P~N π, π 1 π Poit estimate: for μ: μ = x ҧ = 1 σ i=1 x i, with observatio x i R for π: π = p = 1 σ i=1 x i, with observatio x i = 0 or 1 Stadard error of estimate: for μ: σ ത X = σ X for π: σ P = π 1 π Both stadard errors decrease with
ESTIMATING PROPORTION Estimatig π by p ad estimatig σ P = stadard error of proportio So, we have for π a poit estimate p = Y π 1 π by s P = p 1 p a iterval estimate p z α/2 1 α cofidece iterval for π p 1 p, p + z α/2 p 1 p p z α/2 p 1 p π p + z α/2 p 1 p
ESTIMATING PROPORTION Example Cotext: a sample of 75 retail i-store purchases showed that 24 were paid i cash. Give a 95% cofidece iterval for π. p = y = 24 = 0.32; this is the poit estimate for π 75 stadard error of the estimate: s P = p 1 p = 0.32 1 0.32 75 = 0.054 Check validity: p 5 ad 1 p 5 CI π,0.95 : 0.32 1.96 0.054, 0.32 + 1.96 0.054 = 0.214, 0.426 or: 0.214 π 0.426 or: 0.32 ± 0.106
EXERCISE 1 You flip a coi 100 times ad fid 45 times head. Give a 95% cofidece iterval for π head.
HYPOTHESES ON THE PROPORTION Test a hypothesis o the proportio of a Beroulli process Example: you are a police officer you woder if less tha 50% of the (oe-sided) traffic accidets occur with female drivers drivig the car
HYPOTHESES ON THE PROPORTION Statistical model each accidet has a uderlyig Berouilli process of happeig to a ma (0) or to a woma (1), X~alt π you observe the ext = 5 car accidets, ad report the outcomes (0/1) you defie Y as the umber of accidets that is caused by a woma the sequece of 5 observatios ca be regarded as a biomial process, Y~Bi π, 5 you start by assumig the accidet rates are equal, i.e., hypothesize that π = 0.5 Suppose you observed y = 1, i.e., oe car accidet by a woma
HYPOTHESES ON THE PROPORTION Step 1: H 0 : π 0.5; H 1 : π < 0.5; α = 0.05 Step 2: sample statistic: Y =#female; reject for too small values Step 3: if H 0 is just true, Y~Bi 0.5,5 ; o assumptios required Step 4: p value = P Bi 0.5,5 Y 1 = P Y = 0 + P Y = 1 = 0.0313 + 0.1563 = 0.1876 Step 5: p value > α ; do ot reject H 0 ; there is ot sufficiet evidece for cocludig that π < 0.5
HYPOTHESES ON THE PROPORTION What if we have a large sample, say = 100? biomial tables ad formulas do t work Use ormal approximatio if Y~Bi π, the Z = Y π π 1 π ~N 0,1 coditios: π 5 ad 1 π 5: OK Example same as before (car accidets by geder) but ow based o = 100 with y = 40 observed accidets by wome
HYPOTHESES ON THE PROPORTION Step 1: H 0 : π 0.5; H 1 : π < 0.5; α = 0.05 Step 2: sample statistic: Y =#female; reject for too small values Step 3: if H 0 is just true, Z = Y π σ Y = Y π π 1 π ~N 0,1 ormal approximatio OK (π 5 ad 1 π 5) Step 4: z calc = 40 100 0.5 100 0.5 1 0.5 = 2.00 (see, however, ext page!) z crit = 1.645 Step 5: reject H 0, accept H 1 ; there is sufficiet evidece for cocludig that π < 0.5
HYPOTHESES ON THE PROPORTION Note: we forgot about the cotiuity correctio a slightly more accurate result ca be achieved with the cotiuity correctio Example: P X 40 P X 40 1 2 = P Z 401 2 100 0.5 100 0.5 1 0.5 P Z 1.9 < 0.05 Whe eeded? ot whe p value = 0.002 or p value = 0.743 but required i cases like the example, whe p value α =
OLD EXAM QUESTION 21 May 2015, Q1m
FURTHER STUDY Doae & Seward 5/E 11.1-11.2 Tutorial exercises week 5 cofidece itervals hypothesis tests (biomial) hypothesis tests (ormal)