In February 2012, President Obama began to do battle with Catholic Church leader
ID: 3130647 • Letter: I
Question
In February 2012, President Obama began to do battle with Catholic Church leaders over a requirement that health plans cover the cost of birth control. During the initial hubbub, Gallup did a national poll of Catholics to see whether their approval of the president had changed as a result of it. Earlier, the president’s approval rating among Catholics had been 0.49. The survey of 755 Catholics after the announcement gave him an approval rating of 0.46. Compute the 95 percent confidence interval for the survey. Based on the survey results, are we 95 percent certain that Obama’s approval ratings changed? (Is the earlier 0.49 rating within this range?)
Explanation / Answer
Note that
p^ = point estimate of the population proportion = x / n = 0.46
Also, we get the standard error of p, sp:
sp = sqrt[p^ (1 - p^) / n] = 0.01813854
Now, for the critical z,
alpha/2 = 0.025
Thus, z(alpha/2) = 1.959963985
Thus,
Margin of error = z(alpha/2)*sp = 0.035550885
lower bound = p^ - z(alpha/2) * sp = 0.424449115
upper bound = p^ + z(alpha/2) * sp = 0.495550885
Thus, the confidence interval is
( 0.424449115 , 0.495550885 ) [ANSWER, CONFIDENCE INTERVAL]
*********************************
As 0.49 is still within this interval, then NO, there is no significant evidence that Obama's approval ratings has changed. [CONCLUSION]