Question

1, You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable estimate for the population proportion, so we assume p=.5. You would like to be 99% confident that you esimate is within 4% of the true population proportion. How large of a sample size is required?
n =

Hint: Shouldn't the answer be a WHOLE NUMBER.
Do not round mid-calculation. However, use a critical value accurate to three decimal places.

2. You want to obtain a sample to estimate a population proportion. Based on previous evidence, you believe the population proportion is approximately 84%. You would like to be 98% confident that your estimate is within 3.5% of the true population proportion. How large of a sample size is required?

n =

3. A political candidate has asked you to conduct a poll to determine what percentage of people support her, assume p=.5.
If the candidate only wants a 2% margin of error at a 99% confidence level, what size of sample is needed?

4. If n = 540 and ˆp (p-hat) = 0.35, construct a 99% confidence interval.

Give your answers to three decimals

< p <

Answer 1

1)

sample proportion ,   p̂ =    0.5  
sampling error ,    E =   0.04  
Confidence Level ,   CL=   99%  
          
alpha =   1-CL =   1%  
Z value =    Zα/2 =    2.5758   [excel formula =normsinv(α/2) ]
          
Sample Size,n =    (Z / E)²*p̂*(1-p̂)=   1036.7026  
          
          
so,Sample Size required=       1037  

2)

p̂ =    0.84  
sampling error ,    E =   0.035  
Confidence Level ,   CL=   98%  
          
alpha =   1-CL =   2%  
Z value =    Zα/2 =    2.3263   [excel formula =normsinv(α/2) ]
          
Sample Size,n =    (Z / E)²*p̂*(1-p̂)=   593.7621  
          
          
so,Sample Size required=       594  

3)

sample proportion ,   p̂ =    0.5  
sampling error ,    E =   0.02  
Confidence Level ,   CL=   99%  
          
alpha =   1-CL =   1%  
Z value =    Zα/2 =    2.5758   [excel formula =normsinv(α/2) ]
          
Sample Size,n =    (Z / E)²*p̂*(1-p̂)=   4146.8104  
          
          
so,Sample Size required=       4147  
4)

Level of Significance,   α =    0.01
Sample Size,   n =    540
      
Sample Proportion ,    p̂ = x/n =    0.35
z -value =   "Zα/2 = 2.5758 [excel formula =normsinv(α/2) ]
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0205
      
margin of error ,    E = Z*SE =    0.0529
      
Confidence Interval      
Interval Lower Limit , =    p̂ - E =    0.297
Interval Upper Limit , =    p̂ + E =   0.403

answer: 0.297<p<0.403