Question

1. For several years, Democrats have account for 90% of the registered voters in the District. A campaign strategist takes a survey of 75 voters and finds that there are 64 registered Democrats. Has this percentage of registered Democrats changed?

A, state the appropriate hypotheses and test using (alpha = 0.07)

B, Draw the necessary sampling distribution indicating the critical values for (alpha = 0.07)

C, state the decision

D, State the conclusion

E, what error is risked

F construct 93% confidence interval and explain carefully how it can be used to test the hypotheses

Answer 1

a)

Ho :   p =    0.9
H1 :   p ╪   0.9

Number of Items of Interest,   x =   64                  
Sample Size,   n =    75                  
                          
Sample Proportion ,    p̂ = x/n =    0.8533                  
                          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0346                  
Z Test Statistic = ( p̂-p)/SE = (   0.8533   -   0.9   ) /   0.0346   =   -1.3472

b)

sampling distribution

µp = 0.90

σx=Standard Error ,    SE = √( p(1-p)/n ) =    0.0346

critical z value =    ±    1.8119   [excel formula =NORMSINV(α/2)]

c) | test stat |< 1.8119 , fail to reject Ho

d) conclusion: there is not enough evidence to say that this percentage of registered Democrats changed

e) Type II error

f)

Level of Significance,   α =    0.07          
Number of Items of Interest,   x =   64          
Sample Size,   n =    75          
                  
Sample Proportion ,    p̂ = x/n =    0.8533          
z -value =   Zα/2 =    1.812   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0409          
margin of error , E = Z*SE =    1.812   *   0.0409   =   0.0740
                  
93%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.853   -   0.0740   =   0.7793
Interval Upper Limit = p̂ + E =   0.853   +   0.0740   =   0.9274
                  
93%   confidence interval is (   0.7793   < p <    0.9274   )

since, 0.90 is contained in confidence interval, So, fail to reject Ho