Many elementary school students in a school district currently have ear infectio

Question

Many elementary school students in a school district currently have ear infections. A random sample of children in two different schools found that 16 of 42 at one school and 17 of 30 at the other have ear infections. At the level of significance, is there sufficient evidence to support the claim that a difference exists between the proportions of students who have ear infections at the two schools?

No, because the test value -1.56 is inside the noncritical region -1.96 < z < 1.96.

Yes, because the test value -2.04 is outside the noncritical region -1.96 < z < 1.96.

Yes, because the test value -7.33 is outside the noncritical region -1.96 < z < 1.96.

Yes, because the test value -13.09 is outside the noncritical region -1.96 < z < 1.96.

which one its answer?

Explanation / Answer

Formulating the hypotheses          
Ho: p1 - p2   =   0  
Ha: p1 - p2   =/=   0  
Here, we see that pdo =    0   , the hypothesized population proportion difference.  
          
Getting p1^ and p2^,          
          
p1^ = x1/n1 =    0.380952381      
p2 = x2/n2 =    0.566666667      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.117473953      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    -1.580897564 --> -1.56 [closest]

[The z score above is the exact z score. Any deviation is only due to roundoff conventions in your instructor's solution.]

Hence,

OPTION A: No, because the test value -1.56 is inside the noncritical region -1.96 < z < 1.96. [ANSWER, A]  
          

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