Onesample test of proportions


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1 Oesample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples: Drug X is admiistered to 100 patiets with a particular disease. 50 improve. Test whether this drug is better tha drug Y, which is kow to produce improvemet i 45% of patiets. I a poll of 500 voters, 200 say that they will support a particular cadidate i the electio. Give a 95% cofidece iterval for the proportio of all voters who will support the cadidate.
2 Oesample test of proportios Iferetial Procedures: Hypothesis Test  critical value method  pvalue method  cofidece iterval method Cofidece Iterval
3 Oesample test of proportios Hypothesis Test: Specify the ull ad alterative Defie rejectio regio for the test statistic z HA: p p H0: p= p0 < 0 HA: p p0 0 HA: p> p z z z
4 Oesample test of proportios Test Statistic: z = p pˆ If H 0 is true, z comes from stadard ormal distributio Critical value method: Choose z* from ormal table to defie rejectio regio If z falls i rejectio regio, reject H 0 Pvalue method: Use the ormal table to determie probability of a value at least as extreme as z comig from stadard ormal distributio. Report this pvalue ad your coclusio. p 0 (1 p ) 0 0
5 Oesample test of proportios LevelC Cofidece Iterval: ˆ (1 ˆ) p = pˆ ± z* p p To perform a twosided test of H 0 : p = p 0 at sigificace level α, you ca calculate the level(1 α) cofidece iterval ad check whether p 0 falls withi the iterval.
6 Twosample tests: overview Types of twosample tests: twosample ztest twosample ttest twosample test of proportios
7 Twosample ztest The Settig: You wat to determie whether two differet populatios have the same mea for some variable of iterest, so you draw two idepedet samples. Furthermore, you kow the variaces i these two populatios without error. I real life, this would oly ted to happe if your samples were so large (e.g., 5,000) that you could preted that the sample variaces were really the true variaces Example: SAT scores for studets who use a particular prep course have a SD of 100, while those for studets who do ot use the course have a SD of 150. You sample 50 studets who take the course ad 50 who do ot. Test whether takig the course chages the mea score for studets
8 Twosample ztest Hypothesis Test H : μ μ A x < y H μ H μ 0: x y A: x μy = μ H : μ μ A x > y z z z Where the test statistic is z = X σ 2 x x Y σ + 2 y y
9 Twosample ttest The Settig: You wat to determie whether two differet populatios have the same mea value for some variable of iterest. You draw two idepedet samples to test this. Note that i geeral we will use this test istead of the z test, sice we will ot kow the populatio stadard deviatios without error. Example: You wat to determie whether studets who take a give prep course have, o average, the same SAT score as those who do ot. You sample 50 studets who take the course ad 50 who do ot ad record their SAT scores
10 Hypothesis Test Twosample ttest Null ad alterative hypotheses same as for twosample ztest Test statistic is ow t = X s 2 x x Y + s 2 y y If H 0 is true, t follows a tdistributio with df = mi( x 1, y 1)
11 Twosample ttest Commo Mistake: Sometimes you will have what appear to be two idepedet samples, but are really two observatios each o a sigle set of uits. Use oesample methods for iferece o such data. This sample will cosist of the differeces betwee the two values for each subject.
12 Not a Twosample ttest Example: You wat to determie whether a drug lowers patiet cholesterol. You record the cholesterol levels of 100 voluteers, the admiister the drug for three moths. After this period you agai measure the patiets cholesterol values Strategy: Defie D = chage i a patiet s cholesterol after takig drug. Now do a oesample ttest: H0: μ D = 0 HA: μ D< 0 The test statistic will the be t = D s 2 D D 0
13 Twosample test of proportios The Settig: I two idepedet populatios, idividuals ca be classified ito oe of two categories. You draw samples from both populatios to make iferece about whether the category proportios are the same i both populatios Example: You take a sample of 100 govermet cocetrators ad 100 cocetrators from other fields. Test whether beig a govermet cocetrator makes a college studet more or less likely to vote.
14 Twosample test of proportios Hypothesis test: H : p A x y H0: px = py < p HA : px py H : p p A x > y Where z z z The test statistic is pˆ = pˆ x x y y x + pˆ + y z = pˆ x pˆ 1 1 pˆ(1 pˆ) + y x y
15 Twosample test of proportios LevelC Cofidece Iterval: pˆ (1 ˆ x px) ( p ) ( ˆ ˆ x py = px py) ± z* + x pˆ y (1 pˆ ) y y Commo Mistake: Notice that i this case the stadard errors of the hypothesis test ad cofidece iterval are differet!
16 Chisquare test The Settig: I two (or more) idepedet populatios, idividuals ca be classified ito two (or more) differet categories. You draw samples from all populatios to test whether the category a idividual falls ito is idepedet of that idividual s populatio. Example: You choose a sample of 100 me ad 100 wome ad ask subject whether s/he watches professioal football. Test whether geder is idepedet of football viewig
17 Chisquare test Hypothesis test (Cotiuig our example): H 0 : geder is idepedet of watchig football H A : geder is ot idepedet of watchig football Remember that we caot test a oesided alterative usig the chisquare test!
18 Chisquare test Procedure: 1. Create the 2x2 table of observed couts 2. Create the 2x2 table of expected couts 3. Calculate the χ 2 statistic 4. Draw coclusio
19 Chisquare test 1. Create the 2x2 table of observed couts 2. Create the 2x2 table of expected couts Expected cout i a cell = r c Where r = row total for that cell, c = colum total for that cell, = total for whole table
20 Chisquare test 3. Calculate the χ 2 statistic χ ( Observed Expected) 2 = Expected 2 Where the summatio is over all cells.
21 Chisquare test 4. Draw coclusio Uder H 0, χ 2 follows a chisquare distributio with (r1)(c1) degrees of freedom To use the critical value method, determie the value x* that cuts off α of the tail area. Reject H 0 if χ 2 > x*. To use the pvalue method, determie the area to the right of χ 2 o the chisquare curve. Report this pvalue ad draw your coclusio. Area = α Area = pvalue x* χ 2
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