Question

A sample survey contacted an SRS of 2854 registered voters shortly before the 2012 presidential election and asked respondents whom they planned to vote for. Election results show that 51% of registered voters voted for Barack Obama. We will see later that in this situation the proportion of the sample who planned to vote for Barack Obama (call this proportion V) has approximately the Normal distribution with mean μ 0.52 and standard deviation σ = 0.009.

(a) If the respondents answer truthfully, what is P (0.5くV < 0.54)? This is the probability that the sample proportion v estimates the population proportion 0.52 within plus or minus 0.02.
P (0.5<= V <=0.54) (±0.0001)=

(b) In fact, 50% of the respondents said they planned to vote for Barack Obama V = 0.5. If respondents answer truthfully, What is P(V <=0.5)?
P (V <=0.5) (±0.0001) =

Answer 1

a) 97.37%

b) 1.31%

Explanation:

a)

Here we want to calculate the area under the Normal curve with mean 0.52 and standard deviation 0.009 between 0.5 and 0.54

This can be easily done with a spreadsheet and we get

P (0.5くV < 0.54) = 0.9737 or 97.37%

(See picture 1)

b)

Here we want the area under the Normal curve with mean 0.52 and standard deviation 0.009 to the left of 0.5.

P(V ≤ 0.5) = 0.0131 or 1.31%

(See picture 2)