Question

Mars Inc., makers of M & M's candies, claims that they produce plain M & M's with the following distribution Brown: 30% Red: 20% Yellow: 20% Orange: 10% Green: 1096 Blue: 10% A bag of plain M & M's was randomly selected trom the grocery store shelf, and the color counted were as follows Brown: 16 Red: 11Yellow: 19 Orange: 5 Green: 7 Blue: 3 How many total M & Ms were in this sample (This is your n)? What proportion are brown? (This is your p-hatt) You want to conduct an appropriate test of the manufacturer's claim for the proportion of brown M & M's (a) Carry out an appropriate test and state your conclusions (Do not forget to use all PHANTOMS or State, Plan , Do, Conclude) (b) BONUS: Based on this sample, construct and interpret at 95% confidence interval for the proportion of brown M & M 's candies produced by Mars

Answer 1

p = 30% = 0.3, n = 61

(a) Hypothesis test:

Data:    

n = 61   

p = 0.3   

p' = 16/61 = 0.262295082   

Hypotheses:    

Ho: p = 0.3   

Ha: p ≠ 0.3   

Decision Rule:    

α = 0.05   

Lower Critical z- score = -1.9600   

Upper Critical z- score = 1.9600   

Reject Ho if |z| > 1.9600   

Test Statistic:    

SE = √{p (1 - p)/n} = √(0.3 * (1 - 0.3)/61) = 0.0587

z = (p'- p)/SE = (0.262295081967213 - 0.3)/0.0586738694038468 = -0.6426

p- value = 0.5205   

Decision (in terms of the hypotheses):    

Since 0.6426 < 1.9600 we fail to reject Ho

Conclusion (in terms of the problem):    

There is no sufficient evidence that the proportion of brown candies is different from 30%

(b) Confidence interval:

n = 61    

p = 16/61 = 0.262295082    

% = 95    

Standard Error, SE = √{p(1 - p)/n} =    √(0.262295081967213(1 - 0.262295081967213))/61 = 0.056321148

z- score = 1.959963985    

Width of the confidence interval = z * SE =     1.95996398454005 * 0.0563211476374297 = 0.11038742

Lower Limit of the confidence interval = P - width =     0.262295081967213 - 0.110387420937325 = 0.15190766

Upper Limit of the confidence interval = P + width =     0.262295081967213 + 0.110387420937325 = 0.3726825

The confidence interval is [0.152, 0.373]