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## Statistical Methods II

by: Dimitri Torphy

52

0

8

# Statistical Methods II STA 4163

Dimitri Torphy
University of Central Florida
GPA 3.67

Staff

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COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
8
WORDS
KARMA
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## Popular in Statistics

This 8 page Class Notes was uploaded by Dimitri Torphy on Thursday October 22, 2015. The Class Notes belongs to STA 4163 at University of Central Florida taught by Staff in Fall. Since its upload, it has received 52 views. For similar materials see /class/227640/sta-4163-university-of-central-florida in Statistics at University of Central Florida.

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Date Created: 10/22/15
Lecture amp Examples Topic 2 TwoWay Contingency Table In a oneway table we assume the data comes from multinomial distribution with known cell probabilities This lecture we extend our data to a twoway table where the data are classi ed according to two criteria The twoway table for data in Example 141 can be summarized as follows Treatment Relapse No Yes Totals Row Desipramine 14 10 24 Lithium 6 18 24 Placebo 4 2O 24 Totals Column 24 48 72 We call this a 3 X 2 table because it has 3 rows and 2 columns This table shows the relationship between two categorical variables Treatment and Relapse This twoway table gives the observed counts for all 6 combinations of values of these variables Each of the 6 observed counts occupies a cell of the table Row total is the total observed counts in the same row and column total is the total observed counts in the same column One interesting question to ask is quotDo you believe the two classi cation variables are independentquot Suppose we make an assumption that the two classi cation variables are independent then we can compute the estimated expected count for each cell with the following formula Em 2 Row Totall39 Column Total j U Total Sample Size and a x2 test of independence can be performed 0 x2 Test of Independence for TwoWay Contingency Table Data for r x c table Column 1 Column 2 Column c ROW 1 x11 x12 x10 ROW 2 x21 x22 x20 Row r xrl xrz xrc Assumptions 1 The 11 observed counts are a random sample from the population of multinomial distribution with r X c possible outcomes 2 The sample size n is large enough so that for every cell the expected count Enj will be equal to 5 or more Hypothesis H 0 The two classi cations are independent Ha The two classi cations are dependent Test Statistic 2 xijEnij2 X6 Emu Rejection Region at at 2 2 X6 gt XOLr 1c 1 I Example 133 Continuation of Example 131 a Find the estimated expected counts for each cell TwoWay Table Relapse No Yes Row Total Desipramine 14 10 24 Lithium 6 18 24 Placebo 4 2O 24 Column Total 24 48 72 Solution Emu r1ncl 24394 2 8 Engtltrgtczgt lt24g48gt16 47121 18 E 22E72 16 b Suppose we want to test that the treatment and relapse times are independent Write down the hypothesis Solution H 0 Treatment Group and Relapse Response are independent H a Treatment Group and Relapse Response are dependent c Compute the x2 test statistic and nd the critical value at OL 005 Solution xi Enij2 A 14 1 E01 x3 X5905 599147 d Is there enough evidence to say that treatment and relapse are dependent at CC 005 Solution Since we reject the null hypothesis at CC 005 we have enough eVidence to believe there are some dependence among treatment and relapse time I Example 134 To study the export activity of manufacturing rms on Taiwan researchers mailed questionnaires to an SRS of rms in each of ve industries that export many of their products The response rate was only 125 because private companies don39t like to ll out long questionnaires from academic researchers Here are data on the planned sample sizes and the actual number of responses received from each industry Sample size Responses Metal products 18 5 17 Machinery 3 01 3 5 Electrical equipment 5 52 75 Transportation equipment 100 15 Precision instruments 9O 12 If the response rates differ greatly comparisons among the industries may be dif cult a Write down the twoway table Solution Industry Response Yes No Row Total Metal 17 168 1 8 5 Machinery 3 5 266 3 01 Electrical 75 477 552 Transportation 1 5 8 5 100 Precision 12 78 9O Column Total 154 1074 1228 b Do we have enough evidence to say the response rate depends on the industry at CC 005 Solution Hypothesis H0 Response and Industry are independent H a Response and Industry are dependent df5 12 14 critical value at ca 005 is X21005 948773 E 919 m 153 2320 Ebzgma ii 16180 Elalgcmallz mc 3775 EAng m 1374 f 26325 E 2219 m 8151221235 6922 E12 9 9amp9 35312274 48278 Elalgcm a a 1254 9 611516741 E12229 228 8746 512119 m a11534 1129 E1282 g 9 31 7871 2 753 b77232og 8716180gm b271129g 4871g XE PieJ 2320 16180 1129 7871 328 Thus one fails to reject the null hypothesis at 01 005

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