An auto insurance company is conducting a study to examine the relationship betw

Question

An auto insurance company is conducting a study to examine the relationship between the size of a car involved in an accident and the extent of damage caused to the car. Data for a sample of claims are displayed in the table below:

We would like to conduct a chi-square test for independence at the 1% level of significance.

(a) Find all expected cell counts and cell chi-square values in the table.

(b) What is the value of the test statistic for the appropriate test of significance?

(c) What is the P-value for the appropriate test of significance?

(d) What is the appropriate conclusion?

(e) Suppose we had instead conducted the test using the critical value method. What would be the decision rule and the conclusion?

Observed
Expected
Cell Chi-Square Minor    
Damage    Extensive
Damage Write-Off Row Total Compact 21 43    30 94 Midsize 25 51 58 134 Large 36 53   37 126 Column Total 82 147 125 354

Explanation / Answer

a)

Doing an Expected Value Chart,      

b)

b)

Using chi^2 = Sum[(O - E)^2/E],          
          
chi^2 =    7.573327446   [ANSWER]

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c)  
          
With df = (a - 1)(b - 1), where a and b are the number of categories of each variable,          
          
a =    3      
b =    3      
          
df =    4      
          
          
hence, the p value is          
          
P =    0.108518994   [ANSWER]

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d)

As P > 0.01, we   FAIL TO REJECT THE NULL HYPOTHESIS.      
Hence, there is no significant evidence that there is a relationship between the size of a car involved in an accident and the extent of damage caused to the car. [CONCLUSION]

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e)

As df = 4, then at 0.01 level,

chi^2(critical) =    13.27670414

hence, we reject Ho when chi^2 > 13.277. [ANSWER, DECISION RULE]

As chi^2 = 7.5733 < 13.277, we fail to reject Ho [still].

Hence, there is no significant evidence that there is a relationship between the size of a car involved in an accident and the extent of damage caused to the car. [CONCLUSION]
  

21.77401 39.0339 33.19209 31.03955 55.64407 47.31638 29.18644 52.32203 44.49153

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