Question

(12 points) Time magazine reported the result of a telephone poll of 800 adult Americans. The question posed of the Americans who were surveyed was: "Should the federal tax on cigarettes be raised to pay for health care reform?" The results of the survey were: Non-Smokers Smokers Yes No 351 254 41 154 (a) (6 points) Is there sufficient evidence at the α-005 level to conclude that the two population proportions of people that agreed with the tax increase differ for non-smokers and smokers? (b) (6 points) Is there sufficient evidence at the α-0.05 level to conclude that the proportion of people who agreed with the tax increase is lower than 0.5?

Answer 1

	non smoker	smoker	total
yes	351	41	392
no	254	154	408
total	605	195	800

a)

Ho:   p1 - p2 =   0  
Ha:   p1 - p2 ╪   0  
          
first sample size,     n1=   605  
number of successes, sample 1 =     x1=   351  
proportion success of sample 1 , p̂1=   x1/n1=   0.5802  
          
second sample size,     n2 =    195  
number of successes, sample 2 =     x2 =    41  
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.2103
          
difference in sample proportions,    p̂1 - p̂2 =     0.3699

          
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.4900  
          
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.0412  
          
Z-statistic =    (p̂1 - p̂2)/SE =     8.9859  

          
p-value =        0.0000   [excel formula =2*NORMSDIST(z)]
decision :    p-value<α,Reject null hypothesis

so, there is enough evidence to colcude given statement at α=0.05

b)

Ho :   p =    0.5      
H1 :   p <   0.5      
              
Level of Significance,   α =    0.05      
Number of Items of Interest,   x =   392      
Sample Size,   n =    800      
              
Sample Proportion ,    p̂ = x/n =    0.490  

  
              
Standard Error ,    SE = √( p(1-p)/n ) =    0.0177      
              
Z Test Statistic =    Z = ( p̂-p)/SE =    -0.566

  
              
  
p-Value   =   0.2858   [excel function =NORMSDIST(z)]  
Conclusion:     p value>α ,do not reject null hypothesis

so, there is in sufficient evidence to conlcude the given statement at α=0.05