A researcher wishes to estimate with 90 confidence the popul

A researcher wishes to estimate, with

90%

confidence, the population proportion of adults who think the president of their country can control the price of gasoline. Her estimate must be accurate within

3%

of the true proportion.

a) No preliminary estimate is available. Find the minimum sample size needed.

b) Find the minimum sample size needed, using a prior study that found that

46%

of the respondents said they think their president can control the price of gasoline.

c) Compare the results from parts (a) and (b).

(a) What is the minimum sample size needed assuming that no prior information is available?

n=

(Round up to the nearest whole number as needed.)

Solution

a)

If no estimate is available, we assume it is 0.5.

Note that      
      
n = z(alpha/2)^2 p (1 - p) / E^2      
      
where      
      
alpha/2 =    0.05  
As there is no previous estimate for p, we set p = 0.5.      
      
Using a table/technology,      
      
z(alpha/2) =    1.644853627  
      
Also,      
      
E =    0.03  
p =    0.5  
      
Thus,      
      
n =    751.5398484  
      
Rounding up,      
      
n =    752   [ANSWER]

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b)

Note that      
      
n = z(alpha/2)^2 p (1 - p) / E^2      
      
where      
      
alpha/2 =    0.05  
       
      
Using a table/technology,      
      
z(alpha/2) =    1.644853627  
      
Also,      
      
E =    0.03  
p =    0.46  
      
Thus,      
      
n =    746.7299933  
      
Rounding up,      
      
n =    747   [ANSWER]

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c)

Part b is less than the reult of part a. That\'s because part a is the \"worst case\" scenario.

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a) n = 752 [answer]