Chapter 11 Notes

©2009 by S. Gramlich

(updated 4/14/09)

Multinomial Table Hypothesis Testing

! = Important Note

! These Notes are not meant to replace Reading. Read Chapter first.

Multinomial Probability Distribution meets all requirements of Binomial except will have 2 OR MORE outcomes

11-2 Chi-Square Goodness of Fit/Claimed Distribution Hypothesis Tests (for 1-way Contingency Table)

11-3 Chi-Square Hypothesis Test for Indepedence/ Homogeneity (for 2-way Contigency Table)

1) Identify Hypotheses:

11-2

for Goodness of Fit Test

H₀: fits uniform distribution (probalities equal)

H₁: does not fit uniform (at least 1 of probalities not equal)

for Test of Claimed Distribution

Ho: data fits the given distribution.

H₁: data does not conform to the given distribution

11-3

for Test for Indepedence/ Homogeneity

Ho: factors independent or homogeneous.

H₁: factors dependent or not homogeneous

2) Write given (alpha and contingency tables)

calculate k = # of categories; row & column sums; and N = Grand Total/ Sample Size;

r = # of rows, c = # of columns

3) find critical value (cv) = cutoff value for the critical region

11-2: find c_c² from Table A-4 with df = k -1

11-3: find c_c² from Table A-4 with df = (r-1)*(c-1)

4) calculate test statistic (ts) = convert observed scores into a Pearson X² approximation

first calculate Expected Values = E

11-2

for Goodness of Fit: E= N/k

for Claimed Distribution: E= N*p for each given p%

11-3

for Indepedence/Homogeneity: E = (row total) * (column total) / N for that cell

test statistic: Pearson C² approx= S [(O-E)²/E]

! for Goodness of fit: Observed values will change but Expected Value will be constant

! for Claimed, Independence, & Homogeneity: Observed and Expected Values will change with each cell

5) Draw c² curve, label cv, ts & shade critical region.

6) Traditional Method Decision Rule (DR)

if ts is visually inside critical region, reject null.

if ts is visually outside critical region, fail to reject null.

7) find P-value = area under curve corresponding to test statistic

P-val will be range of values in top row of Table A-4

8) P-Value Method DR

if p-val <= alpha, Reject null.

if p-val > alpha, Fail to Reject null.

9) State conclusion in words relative to the original claim.

TECHNOLOGY

using StatCrunch:

to find cv c_c²: Stat – Calculators – Chi-Square – (enter df , >=, enter α in last box, leave 3rd box blank) – Compute

to find P-value: Stat – Calculators – Chi-Square – (enter df , >=, enter ts in 3^rd box, leave 4^th box blank) – Compute

for Good of Fit/ Claimed Distribution HT:

Enter the Observed values in 1 column and the Expected values in another

Stat – Goodness-of-fit – Chi-Square test – (select Observed & Expected columns) – Calculate

for Test of Independence/ Homogeneity HT:

Enter row variable labels in column 1 and enter column values in rest of columns.

! Do not enter variable labels in 1^st row as in excel.

Stat – Tables – Contigency – with summary

Click “?” for more help and an example on this procedure.

EXCEL commands:

Find Excel Command

cv c_c² =CHIINV(alpha, df)

P-value =CHIDIST(ts, df)

Or =CHITEST(highlight observed, highlight expected)