Consider the following hypotheses Which of the following sam

Consider the following hypotheses:

Which of the following sample information enables us to reject the null hypothesis at = 0.01 and at = 0.05?

  a. x = 35; n = 125 Options are: reject or do not reject for 0.01 and 0.05

b. x = 66; n = 285 Options are: reject or do not reject for 0.01 and 0.05

Solution

a)

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   >=   0.33
Ha:   p   <   0.33
As we see, the hypothesized po =   0.33      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.28      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.042057104      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    -1.188859793      
          
As this is a    1   tailed test, then, getting the p value,  
          
p =    0.117247422      

Thus, we DO NOT REJECT for both 0.01 and 0.05. [ANSWER]

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b)

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   >=   0.33
Ha:   p   <   0.33
As we see, the hypothesized po =   0.33      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.231578947      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.027852998      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    -3.533589244      
          
As this is a    1   tailed test, then, getting the p value,  
          
p =    0.000204979      

Thus, we REJECT for both 0.01 and 0.05. [ANSWER]

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c)

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   >=   0.33
Ha:   p   <   0.33
As we see, the hypothesized po =   0.33      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.3      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.070095173      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    -0.427989525      
          
As this is a    1   tailed test, then, getting the p value,  
          
p =    0.334329373      

Thus, we DO NOT REJECT for both 0.01 and 0.05. [ANSWER]

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d)

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   >=   0.33
Ha:   p   <   0.33
As we see, the hypothesized po =   0.33      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.3      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.022835548      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    -1.313741175      
          
As this is a    1   tailed test, then, getting the p value,  
          
p =    0.09446666      

Thus, we DO NOT REJECT for both 0.01 and 0.05. [ANSWER]