6.14 The Civil War. A national survey conducted in 2011 among a simple random sample of 1,507 adults shows that 56% of Americans think the Civil War is still relevant to American politics and political life.

  1. Conduct a hypothesis test to determine if these data provide strong evidence that the majority of the Americans think the Civil War is still relevant.

\[H_0 :p =0.50\] (50% of Americans think the Civil War is still relevant) \[H_a :p >0.50\] (More than 50% of Americans think the Civil War is still relevant)

Independence Test

1. The sample is random 
2. Sample is less than 10% of US adults
relp0 = 0.56
p0 = 0.5
n = 1507

# Success - failure condition
condition <- n*p0 > 10
condition
## [1] TRUE

Since the observations are independent and the success failure condition is met, it is approximately normal

# compute Z-score
z = (relp0 -p0)/sqrt(p0*(1-p0)/n)
z
## [1] 4.658412
1-pnorm(z,0,1)
## [1] 1.593292e-06

Because the p-value is smaller than 0.05, we reject \(H_0\) The survey provides strong evidence that the majority of Americans think the Civil War is still relevant.

  1. Interpret the p-value in this context. The probability of obtaining a random sample of 1,507 Americans where 50% think the Civil War is irrelevant is approximately zero.

  2. Calculate a 90% confidence interval for the proportion of Americans who think the Civil War is still relevant. Interpret the interval in this context, and comment on whether or not the confidence interval agrees with the conclusion of the hypothesis test.

relp0 = 0.56
n = 1507
U = relp0 + 1.65*sqrt(relp0*(1-relp0)/n)
L = relp0 - 1.65*sqrt(relp0*(1-relp0)/n)
c(L,U)
## [1] 0.5389017 0.5810983

We are confident that 54% to 58% of all Americans think the Civil War is relevant.