Question

Suppose a sample of 50 Americans is taken and they are asked whether they believe the federal government should make all public universities tuition-free. 40% of the sample says yes.

a)Show mathematically that the Central Limit Theorem applies.

b)A local politician claims that over half of Americans support this proposal. Comment on the validity of this statement, based on the 90% Confidence Interval.

Answer 1

np = 20 > 5

and n(1-p)>5

npq>10

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Level of Significance,   α =    0.10          
Number of Items of Interest,   x =   20          
Sample Size,   n =    50          
                  
Sample Proportion ,    p̂ = x/n =    0.400          
z -value =   Zα/2 =    1.645   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0693          
margin of error , E = Z*SE =    1.645   *   0.0693   =   0.1140
                  
90%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.400   -   0.1140   =   0.2860
Interval Upper Limit = p̂ + E =   0.400   +   0.1140   =   0.5140
                  
90%   confidence interval is (   0.2860   < p <    0.5140   )

0.5 lie in the interval, so, claim is not supported

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