A March 2012 Gallup poll asked American adults whether they

A March 2012 Gallup poll asked American adults whether they thought the news media exaggerate the seriousness of global warming. The proportion who responded \"No\" was 0.6. Is this good evidence that the majority of American adults do not believe the news media exaggerate the seriousness of global warming? The answer depends on the size of the sample. Let p denote the true proportion of American adults who do not believe the news media exaggerate the seriousness of global warming.

Give a 95% confidence interval for the true proportion p for three different sample sizes.

Use pˆ=0.6 in all three tests. But use three different sample sizes, n=25, n=100 and n=400.

Solution

Confidence Interval For Proportion
CI = p ± Z a/2 Sqrt(p*(1-p)/n)))
x = Mean
n = Sample Size
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
a)
When n=25
Sample Size(n)=25
Sample proportion = x/n =0.6
Confidence Interval = [ 0.6 ±Z a/2 ( Sqrt ( 0.6*0.4) /25)]
= [ 0.6 - 1.96* Sqrt(0.01) , 0.6 + 1.96* Sqrt(0.01) ]
= [ 0.408,0.792]

b)
When n= 100
Sample Size(n)=100
Confidence Interval = [ 0.6 ±Z a/2 ( Sqrt ( 0.6*0.4) /100)]
= [ 0.6 - 1.96* Sqrt(0.002) , 0.6 + 1.96* Sqrt(0.002) ]
= [ 0.504,0.696]

c)
When n = 400
Sample Size(n)=400
Sample proportion = x/n =0.6
Confidence Interval = [ 0.6 ±Z a/2 ( Sqrt ( 0.6*0.4) /400)]
= [ 0.6 - 1.96* Sqrt(0.001) , 0.6 + 1.96* Sqrt(0.001) ]
= [ 0.552,0.648]