Question

An organization surveyed 613 high school seniors from a certain country and found that 325 believed they would not have enough money to live comfortably in college. The folks at the organization want to know if this represents sufficient evidence to conclude a majority​ (more than 50​%) of high school seniors in the country believe they will not have enough money in college. ​(a) What does it mean to make a Type II error for this​ test? ​(b) If the researcher decides to test this hypothesis at the alpha equals 0.05 level of​ significance, compute the probability of making a Type II​ error, beta​, if the true population proportion is 0.55. What is the power of the​ test? ​(c) Redo part​ (b) if the true population proportion is 0.57.

Answer 1

a)

Type II is conlcuding that  majority​ (more than 50​%) of high school seniors in the country do not believe they will not have enough money in college , while in actual majority​ (more than 50​%) of high school seniors in the country they believe

b) true proportion,   p=   0.55                      
                              
hypothesis proportion,   po=    0.5                      
significance level,   α =    0.050                      
sample size,   n =   613                      
                              
std error of sampling distribution,   σpo = √(po*(1-po)/n) = √ (   0.500   *   0.500   /   613   ) =   0.0202
std error of true proportion,   σp = √(p(1-p)/n) = √ (   0.55   *   0.45   /   613   ) =   0.0201

Zα =       1.645   (right tailed test)      
                  
We will fail to reject the null (commit a Type II error) if we get a Z statistic <                   1.645
this Z-critical value corresponds to X critical value( X critical), such that                  
                  
(p^ - po)/σpo ≤ Zα                  
p^ ≤ Zα*σpo + po                  
p^ ≤    1.645*0.0202+0.5       =   0.5332  
                  
now, type II error is ,ß =    P( p^ ≤    0.5332   given that p =   0.55  
                  
   = P ( Z < (p^ - p)/σp )=       P(Z < (0.5332-0.55) / 0.0201)      
   = P ( Z < (   -0.835   )      
ß   =   0.201798          
power =    1 - ß =   0.7982

c)

true proportion,   p=   0.57                      
                              
hypothesis proportion,   po=    0.5                      
significance level,   α =    0.050                      
sample size,   n =   613                      
                              
std error of sampling distribution,   σpo = √(po*(1-po)/n) = √ (   0.500   *   0.500   /   613   ) =   0.0202
std error of true proportion,   σp = √(p(1-p)/n) = √ (   0.57   *   0.43   /   613   ) =   0.0200

Zα =       1.645   (right tailed test)      
                  
We will fail to reject the null (commit a Type II error) if we get a Z statistic <                   1.645
this Z-critical value corresponds to X critical value( X critical), such that                  
                  
(p^ - po)/σpo ≤ Zα                  
p^ ≤ Zα*σpo + po                  
p^ ≤    1.645*0.0202+0.5       =   0.5332  
                  
now, type II error is ,ß =    P( p^ ≤    0.5332   given that p =   0.57  
                  
   = P ( Z < (p^ - p)/σp )=       P(Z < (0.5332-0.57) / 0.02)      
   = P ( Z < (   -1.839   )      
ß   =   0.032921          
                  
                  
                  
power =    1 - ß =   0.9671