Question

Problem: Proportion of "Cured” Cancer Patients: How Does Canada Compare with Europe? Lung cancer remains the leading cause of cancer death for both Canadian men and women, responsible for the most potential years of life lost to cancer. Lung cancer alone accounts for 28% of all cancer deaths in Canada (32%. in Quebec). Most forms of lung cancer start insidiously and produce no apparent symptoms until they are too far advanced. Consequently, the chances of being cured of lung cancer are not very promising, with the five-year survival rate being less than 15%. The overall data for Europe show that the number of patients who are considered "cured” is rising steadily. For lung cancer, this proportion rose from 6% to 8%. However, there was a wide variation in the proportion of patients cured in individual European countries. For instance, the study shows that for lung cancer, less than 5% of patients were cured in Denmark, the Czech Republic, and Poland, whereas more than 10% of patients were cured in Spain. 1. Suppose a sample of 75 Quebecers was selected and it was found that 27 of them had died due to lung cancer. (a) Perform the appropriate test of hypothesis to determine whether the proportion of deaths in Quebec due to the leading cause (lung cancer) has changed. Test using a = 0.05. (b) Find the p-value for this test and interpret it. (c) Construct a 95% confidence interval estimate of the population proportion of the premature cancer death in Quebec. (d) Explain how to use this confidence interval to test the hypotheses. 2. Suppose two independent samples were taken. The following data were recorded: Quebec: Rest of Canada: ni = 150 n2 = 1000 Number of deaths due to cancer= x1 = 47 Number of deaths due to cancer= x2 = 291

Answer 1

1)

a)

Ho :   p =    0.32  
H1 :   p ╪   0.32   (Two tail test)
          
Level of Significance,   α =    0.05  
Number of Items of Interest,   x =   27  
Sample Size,   n =    75  
          
Sample Proportion ,    p̂ = x/n =    0.3600  
          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0539  
Z Test Statistic = ( p̂-p)/SE =    (0.36-0.32)/0.0539=   0.74  
          
b)
          
p-Value   =   0.4577   [excel formula =2*NORMSDIST(z)]
Decision:   p value>α ,do not reject null hypothesis       
There is not enough evidence that proportion has been changed

c)

Level of Significance,   α =    0.05          
Number of Items of Interest,   x =   27          
Sample Size,   n =    75          
                  
Sample Proportion ,    p̂ = x/n =    0.3600          
z -value =   Zα/2 =    1.960   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.055426          
margin of error , E = Z*SE =    1.960   *   0.05543   =   0.1086
                  
95%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.36000   -   0.10863   =   0.2514
Interval Upper Limit = p̂ + E =   0.36000   +   0.10863   =   0.4686
                  
95%   confidence interval is (   0.251   < p <    0.469   )

d)

Since the confidence interval does include 0.32 which means we can not reject the hypthesis.

Please let me know in case of any doubt.

Thanks in advance!

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