Question

In 1994, 52% of parents with children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 368 of 800 parents with children in high school felt it was a serious problem that high school student were not being taught enough math and science. Do parents feel differently today than they did in 1994?

a. What does making a Type II error for this test mean in context?

b. If the researcher decides to test this hypothesis at the 0.05 level of significance, determine the probability of making a Type II error if the true population proportion is 0.50. What is the power of the test? Sketch the relevant distribution and region.

Answer 1

a) Type II error is concluding that  parents did not feel differently today than they did in 1994, while in actual parents feel differently today than they did in 1994

b)

true proportion,   p=   0.5                      
                              
hypothesis proportion,   po=    0.52                      
significance level,   α =    0.050                      
sample size,   n =   800                      
                              
std error of sampling distribution,   σpo = √(po*(1-po)/n) = √ (   0.520   *   0.480   /   800   ) =   0.0177
std error of true proportion,   σp = √(p(1-p)/n) = √ (   0.5   *   0.5   /   800   ) =   0.0177
Zα/2   = ±   1.960   (two tailed test)                  
We will fail to reject the null (commit a Type II error) if we get a Z statistic between                       -1.960   and   1.960
these Z-critical value corresponds to some X critical values ( X critical), such that                              
-1.960   ≤(p^ - po)/σpo≤   1.960                      
-1.960   *σpo + po≤ p^ ≤   1.960   *σpo + po                  
0.4854   ≤ p^ ≤   0.5546                      
                              
now, type II error is ,ß =        P(0.4854< p^ < 0.5546)       =P( (0.4854-p) /σp < Z < (0.5546-p)/σp )              
       =P( (0.4854-0.5)/0.0177) < Z < (0.5546-0.5)/0.0177 )                      
so, P(   -0.827   < Z <   3.090   ) = P ( Z ≤   3.090   ) - P ( Z ≤   -0.827   )
       =   0.999   -   0.204   =   0.7949  
                              
power =    1 - ß =   0.2051