Question

Suppose that my favorite colors are RED and PURPLE, and I hate all other colors of Skittles. Find a 95% confidence interval for the difference in proportion of Skittles packages whose dominant color is a FAVORITE COLOR and the proportion of Skittles packages whose dominant color is a HATED COLOR.

Package	Red	Orange	Yellow	Green	Purple	Total	Dominant Color
1	15	15	16	12	14	72	Y
2	13	6	11	16	15	61	G
3	9	9	10	18	15	61	G
4	7	9	8	18	15	57	G
5	13	9	14	10	7	53	Y
6	15	20	8	10	9	62	O
7	12	11	11	16	8	58	G
8	14	14	15	5	12	60	Y
9	14	14	15	5	12	60	Y
10	16	10	13	10	10	59	R
11	14	11	12	13	9	59	R
12	10	11	16	8	15	60	Y
13	7	15	12	12	13	59	O
14	7	15	12	12	13	59	O
15	18	10	10	9	13	60	R
16	10	9	12	15	9	55	G
17	8	13	11	6	21	59	P
18	12	14	12	10	11	59	O
19	12	8	11	15	12	58	G
20	14	5	13	14	13	59	R
21	16	10	11	15	10	62	R
22	10	9	9	9	21	58	P
23	15	6	12	12	12	57	R
24	10	12	17	9	8	56	Y
25	16	7	16	10	11	60	R
26	12	10	12	8	16	58	P
27	7	13	5	11	15	51	P
28	8	11	18	12	11	60	Y
29	14	9	10	11	13	57	R
30	12	12	12	7	14	57	P
31	16	7	10	7	13	53	R
32	13	14	13	7	12	59	O
33	10	9	9	9	21	58	P
34	13	7	14	14	10	58	Y

Answer 1

sample #1   -----> fav color
first sample size,     n1=   34          
number of successes, sample 1 =     x1=   15          
proportion success of sample 1 , p̂1=   x1/n1=   0.4412          
                  
sample #2   ----->   hated color   
second sample size,     n2 =    34          
number of successes, sample 2 =     x2 =    19          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.559          
                  
difference in sample proportions, p̂1 - p̂2 =     0.4412   -   0.5588   =   -0.1176

level of significance, α =   0.05              
Z critical value =   Z α/2 =    1.960   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.1204          
margin of error , E = Z*SE =    1.960   *   0.1204   =   0.2360
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.118   -   0.2360   =   -0.3537
upper limit = (p̂1 - p̂2) + E =    -0.118   +   0.2360   =   0.1184
                  
so, confidence interval is (   -0.3537   < p1 - p2 <   0.1184   )  

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