Question

A political pollster is conducting an analysis of sample results in order to make predictions on election night. Assuming a​ two-candidate election, if a specific candidate receives at least 54% of the vote in the​ sample, that candidate will be forecast as the winner of the election. You select a random sample of 100 voters. Complete parts​ (a) through​ (c) below.

a.

The probability is ______ that a candidate will be forecast as the winner when the population percentage of her vote is 50.1​%.

b.

The probability is _____that a candidate will be forecast as the winner when the population percentage of her vote is 56​%

c.

What is the probability that a candidate will be forecast as the winner when the population percentage of her vote is 49​% ​(and she will actually lose the​ election)? d. The probability is _____ that a candidate will be forecast as the winner when the population percentage of her vote is 50.1​%. The probability is_____that a candidate will be forecast as the winner when the population percentage of her vote is 56​%. The probability is ______that a candidate will be forecast as the winner when the population percentage of her vote is 49​%. E. Choose the correct answer below. A. Increasing the sample size by a factor of 4 increases the standard error by a factor of 2. Changing the standard error doubles the magnitude of the standardized​ Z-value. B. Increasing the sample size by a factor of 4 decreases the standard error by a factor of 2. Changing the standard error decreases the standardized​ Z-value to half of its original value. C. Increasing the sample size by a factor of 4 increases the standard error by a factor of 2. Changing the standard error decreases the standardized​ Z-value to half of its original value. D. Increasing the sample size by a factor of 4 decreases the standard error by a factor of 2. Changing the standard error doubles the magnitude of the standardized​ Z-value.

Answer 1

a)

population proportion ,p=   0.501      
n=   100      
          
std error , SE = √( p(1-p)/n ) =    0.0500      
          
sample proportion , p̂ =   0.54      
Z=( p̂ - p )/SE=   0.780      
P ( p̂ >    0.54   ) =P(Z > ( p̂ - p )/SE) =  
          
=P(Z >   0.780   ) =    0.2177(answer)

The probability is _21.77%____ that a candidate will be forecast as the winner when the population percentage of her vote is 50.1​%

b)

population proportion ,p=   0.56      
n=   100      
          
std error , SE = √( p(1-p)/n ) =    0.0496      
          
sample proportion , p̂ =   0.54      
Z=( p̂ - p )/SE=   -0.403      
P ( p̂ >    0.54   ) =P(Z > ( p̂ - p )/SE) =  
          
=P(Z >   -0.403   ) =    0.6565(answer)

The probability is__65.65%___that a candidate will be forecast as the winner when the population percentage of her vote is 56​%.

c)

population proportion ,p=   0.49      
n=   100      
          
std error , SE = √( p(1-p)/n ) =    0.0500      
          
sample proportion , p̂ =   0.54      
Z=( p̂ - p )/SE=   1.000      
P ( p̂ >    0.54   ) =P(Z > ( p̂ - p )/SE) =  
          
=P(Z >   1.000   ) =    0.1586

The probability is ___15.86%___that a candidate will be forecast as the winner when the population percentage of her vote is 49​%

E. Choose the correct answer below

std error , SE = √( p(1-p)/n ) so, SE is inversely proportional sqrt of n

and Z=( p̂ - p )/SE, so, Z is inversely proportional to SE

hence answer is

option d)

Increasing the sample size by a factor of 4 decreases the standard error by a factor of 2. Changing the standard error doubles the magnitude of the standardized​ Z-value.