One-sample test of proportions

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1 Oe-sample 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 Oe-sample test of proportios Iferetial Procedures: Hypothesis Test - critical value method - p-value method - cofidece iterval method Cofidece Iterval

3 Oe-sample 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 Oe-sample 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 P-value method: Use the ormal table to determie probability of a value at least as extreme as z comig from stadard ormal distributio. Report this p-value ad your coclusio. p 0 (1 p ) 0 0

5 Oe-sample test of proportios Level-C Cofidece Iterval: ˆ (1 ˆ) p = pˆ ± z* p p To perform a two-sided 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 Two-sample tests: overview Types of two-sample tests: two-sample z-test two-sample t-test two-sample test of proportios

7 Two-sample z-test 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 Two-sample z-test 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 Two-sample t-test 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 Two-sample t-test Null ad alterative hypotheses same as for two-sample z-test Test statistic is ow t = X s 2 x x Y + s 2 y y If H 0 is true, t follows a t-distributio with df = mi( x 1, y 1)

11 Two-sample t-test 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 oe-sample methods for iferece o such data. This sample will cosist of the differeces betwee the two values for each subject.

12 Not a Two-sample t-test 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 oe-sample t-test: H0: μ D = 0 HA: μ D< 0 The test statistic will the be t = D s 2 D D 0

13 Two-sample 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 Two-sample 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 Two-sample test of proportios Level-C 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 Chi-square 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 Chi-square 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 oe-sided alterative usig the chi-square test!

18 Chi-square 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 Chi-square 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 Chi-square test 3. Calculate the χ 2 statistic χ ( Observed Expected) 2 = Expected 2 Where the summatio is over all cells.

21 Chi-square test 4. Draw coclusio Uder H 0, χ 2 follows a chi-square distributio with (r-1)(c-1) 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 p-value method, determie the area to the right of χ 2 o the chi-square curve. Report this p-value ad draw your coclusio. Area = α Area = p-value x* χ 2

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