Question

2. Below is some data on personality type (introverted or extroverted) and favourite colour. a) Run the appropriate test at a 5% significance level to determine if there is a relationship between these variables. (Be sure to state the hypotheses, test statistic, p-value, and conclusions.)

Run the standardized residuals. Provide the numbers and an interpretation.

	Colours
	Red	Yellow	Green	Blue
Introvert personality	20	6	30	44
Extrovert personality	180	34	50	36

Answer 1

Ho: given two variable are independent
H1: Given two variables are not independent

Chi-Square Test of independence
	Observed Frequencies
	0
0	red	yellow	green	blue			Total
introvert	20	6	30	44			100
extrovert	180	34	50	36			300
Total	200	40	80	80			400
	Expected frequency of a cell = sum of row*sum of column / total sum
	Expected Frequencies
	red	yellow	green	blue			Total
introvert	200*100/400=50	40*100/400=10	80*100/400=20	80*100/400=20			100
extrovert	200*300/400=150	40*300/400=30	80*300/400=60	80*300/400=60			300
Total	200	40	80	80			400
	(fo-fe)^2/fe
introvert	18.000	1.600	5.000	28.800
extrovert	6.000	0.533	1.667	9.600

Chi-Square Test Statistic,χ² = Σ(fo-fe)^2/fe =   71.200  
      
Level of Significance =   0.01  
Number of Rows =   2  
Number of Columns =   4  
Degrees of Freedom=(#row - 1)(#column -1) = (2- 1 ) * ( 4- 1 ) =   3  
      
p-Value =   0.0000   [Excel function: =CHISQ.DIST.RT(χ²,df) ]
Decision:    p-value < α , Reject Ho  
conclusion: there is a relationship between these variables

........................

Please revert back in case of any doubt.

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