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LECTURE 10 TESTS FOR PROPORTIONS

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EPSY 640

Texas A&M University

- Proportions involve Nominal scale data
- Univariate case
- 1 sample
- k samples

- Bivariate case
- association
- agreement

- Proportion = k/n where k= # in category of interest, n=sample size
- Var (p) = p(1-p)/n = pq/n, where p= proportion, n= # in sample, andq=1-p
- If you have a single case, then Var(p)=p(1-p) = pq
- ex. Single roll of two dice, getting a 7 has p=1/6, so for a single roll, Var(p)=1/6*5/6 = 5/36, SD=sqrt(5/36) = .373

- The confidence interval due to Ghosh is
- p [n/(n+z2){ p + z2/2n z pq/n + z2/rn2}
- ex. The probability of admission to med school is .08 for applicants. The 95% confidence interval around a single case is(.01, .81)
- For the roll of a 7, the 95% CI is (.005, .846)

Excel formulas for confidence interval around a single proportion

- For k categories with pi = proportion in category i and p1+p2+...pk=1, we can test 1. All proportions are the same2. Each pi = i based on theory or previous data

- For k categories with pi = proportion in category i and p1+p2+...pk=1
- All proportions are the same p.=1/k:
X2 = N[(p1 - p.)2/p.+ (p2-p.)2/p.+...(pk-p.)2/p.]

with degrees of freedom k-1

ex. The reported proportions of ethnic groups in Bryan is .33 Caucasian, .42 Hispanic, and .25 African American with 8000 students. The chi square is 347.2 with 2df. This is significant at p=.0001

- For k categories with pi = proportion in category i and p1+p2+...pk=1, each proportion tested against a theoretical value i
X2 = N[(p1 - 1)2/ 1+(p2- 2)2/ 2+...(pk- k)2/k]

with degrees of freedom k-1

ex. The reported proportions of ethnic groups in Bryan is .33 Caucasian, .42 Hispanic, and .25 African American with 8000 students. National proportions are .74, .14, and .12.

The chi square is 7423.9 with 2df. This is significant at p=.0001

- Two nominal variables are measured for N persons (eg. Gender and aggression status)
- An R x C table of proportions is computedprc = proportion for row r, column c
- Then, the sum of a row of proportions is pr. and for a column the sum is p.c
- Association is defined as departure from average expected proportion for each cell cumulated over the cells

- For an R x C table of proportions with the sum of all cell proportions equal to 1.0,
X2 = N[(p11 - p1.p.1)2/ p1.p.1 +(p12- p1.p.2 )2/ p1.p.2 +...(prc- pr.p.c )2/ pr.p.c]

with (R-1)(C-1) degrees of freedom

- For an R x C table of proportions with the sum of all cell proportions equal to 1.0,
X2 = SUM [ (Oij – Eij ]2 / Eij

Eij = expected cell count based on row and column averages, = N*p.j*pi.

16

12

22

50

11

23

10

44

27 35 32

OBSERVED CELLS

DEM REP IND

M

F

X2(2,.05)=5.99

X2 = zi2, where zi= z-score for score I

TAAS PASS %

ETHNIC %

TAAS PASS N

TAAS PASS PROPORTIONS

Question: Is school passing rate related to ethnicity?

TAAS % PASSING

N=840

Expected %

(TAAS% - Expected)2 /Expected

Chi Square statistic

significant p< .001

Conclusion: school is associate with ethnic TAAS pass rate