Question

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The Arizona Department of Transportation (ADOT) is considering changing statewide speed limits from 65 mph to 75 mph. They wish to perform analysis on p, the proportion of Arizona residents that favor this change. In order to prove that a majority of residents favor the change, ADOT wishes to test the following hypotheses using a level of significance of 0.05. Ho:p=0.5 Hp > 0.5 ADOT wishes to design the test to have a power of 90% if the true population proportion is equal to 0.55.

Answer 1

Ho :   µ =   0.5              
Ha :   µ >   0.5              
                      
true proportion   p =    0.55              
hypothesis proportion,   po =    0.5              
                      
Level of Significance ,    α =    0.05              
power =    1-ß =    0.90              
ß=       0.10              
                      
                      
Z ( α ) =       1.645              
           [excel function: =normsinv(α) ]           
Z (ß) =        1.282             
           [excel function: =normsinv(ß)         
sample size needed =    n = po*(1-po)[Z(α) + Z(ß) ]² / (p-po)² =   856.3847              
                      
so,Required sample size =        857              

b)

Number of Items of Interest,   x =   461
Sample Size,   n =    857
      
Sample Proportion ,    p̂ = x/n =    0.5379

Please let me know in case of any doubt.

Thanks in advance!

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