Question

Problem 1 (Test for the difference among multiple proportions)

How common are financial cost or contractual constrains associated with smartphone ownership? A survey of smartphone owners found that 48% of the 18 – to 29 –years–olds, 38% of the 30- to 49-years-olds, 25% of the 50- to 64-years-olds, and 19% of those 65 or older have reached the maximum amount of data they are allowed to use as part of their plan, at least on occasion.

Suppose the survey was based on 200 smartphone owners in each of the four age groups: 18 to 29, 30 to 49, 50 to 64, and 65 +.

1. At the 0.05 level of significance, is there evidence of a difference among the age groups in the proportion of smartphone owners who have reached the maximum amount of data they are allowed to use as part of their plan, at least on occasion?

Problem 2 (Test for independence among proportions)

What makes sales leaders tick? Mercuri International conducted a study to explore sales strategies, processes, and support systems within businesses. Organizations were categorized by sales performance level (top performers vs. middle performers vs. bottom performers) and extent to which the organization invests in customer satisfaction. Results were organized in the following table.

	Sales Performance Level
Level Of Investment	Top	Middle	Bottom	Total
Annually	53	318	44	415
Every 2-4 years	40	245	23	308
Never	11	158	34	203
Total	104	721	101	926

At the 0.05 level of significance, is there evidence of a significant relationship between sales performance level and level of investment in customer satisfaction?

Answer 1

1)

observed frequencey, O	expected proportion	expected frequency,E	(O-E)		(O-E)²/E
96	0.250	65.00	31.00		14.785
76	0.250	65.00	11.00		1.862
50	0.250	65.00	-15.00		3.462
38	0.250	65.00	-27.00		11.215

chi square test statistic,X² = Σ(O-E)²/E =   31.323              
                  
level of significance, α=   0.05              
Degree of freedom=k-1=   4   -   1   =   3
                  
P value =   0.0000   [ excel function: =chisq.dist.rt(test-stat,df) ]          
Decision: P value < α, Reject Ho  

There is enough evidence of     difference among the age groups in the proportion of smartphone owners who have reached the maximum amount of data they are allowed to use as part of their plan, at least on occasion   
===============

(Test for independence among proportions)

Ho: There is no significant relationship between sales performance level and level of investment in customer satisfaction

Ha: There is a  significant relationship between sales performance level and level of investment in customer satisfaction

Chi-Square Test of independence
	Observed Frequencies
Level Of Investment	Top	Middle	Bottom				Total
Annually	53	318	44				415
Every 2-4 years	40	245	23				308
Never	11	158	34				203
Total	104	721	101				926
	Expected frequency of a cell = sum of row*sum of column / total sum
	Expected Frequencies
	Top	Middle	Bottom				Total
Annually	104*415/926=46.609	721*415/926=323.126	101*415/926=45.265				415
Every 2-4 years	104*308/926=34.592	721*308/926=239.814	101*308/926=33.594				308
Never	104*203/926=22.799	721*203/926=158.059	101*203/926=22.141				203
Total	104	721	101				926
(fo-fe)^2/fe	Top	Middle	Bottom				total
Annually	0.876	0.081	0.035				0.993
Every 2-4 years	0.846	0.112	3.341				4.299
Never	6.106	0.000	6.351				12.458

Chi-Square Test Statistic,χ² = Σ(fo-fe)^2/fe =   17.749  
      
Level of Significance =   0.05  
Number of Rows =   3  
Number of Columns =   3  
Degrees of Freedom=(#row - 1)(#column -1) = (3- 1 ) * ( 3- 1 ) =   4  
      
p-Value =   0.001   [Excel function: =CHISQ.DIST.RT(χ²,df) ]
Decision:    p-value < α , Reject Ho  

there is a evidence of a significant relationship between sales performance level and level of investment in customer satisfaction