Question

Many news organizations conduct polls asking adults in the United Stats in they approve of the job the president is doing. According to a Gallup poll of 1500 randomly selected U.S. adults in October 2012, 780 approved of President Obama's job performance. A Gallup poll of 1500 randomly selected U.S. adults in October 2014 showed that 615 approved of Obama's job performance. (a) Construct a 90% confidence interval for the change in Obama's approval rating among U.S. adults from October 2012 to October 2014. (b) Does the interval give evidence that his approval rating increased, decreased, or did not change? Justify your answer.

Answer 1

To construct a 90% confidence interval for the change in Obama's approval rating from October 2012 to October 2014, we use the formula CI = p2 - p1 ± Z * sqrt((p2(1 - p2)/n2) + (p1(1 - p1)/n1)). Calculating the confidence interval, we find that the change in Obama's approval rating is between -0.18538 and -0.03462. This provides evidence that his approval rating decreased.

Step-by-step explanation:

To construct a 90% confidence interval for the change in Obama's approval rating among U.S. adults from October 2012 to October 2014, we can use the formula:

CI = p2 - p1 ± Z * sqrt((p2(1 - p2)/n2) + (p1(1 - p1)/n1))

where:

• p1 = proportion who approved in 2012 (780/1500)
• p2 = proportion who approved in 2014 (615/1500)
• Z = Z-score for a 90% confidence interval (1.645)
• n1 = sample size in 2012 (1500)
• n2 = sample size in 2014 (1500)

Calculating the confidence interval:

p1 = 780/1500 = 0.52

p2 = 615/1500 = 0.41

CI = 0.41 - 0.52 ± 1.645 * sqrt((0.41(1 - 0.41)/1500) + (0.52(1 - 0.52)/1500))

CI = -0.11 ± 1.645 * sqrt(0.000976 + 0.001139)

CI = -0.11 ± 1.645 * sqrt(0.002115)

CI = -0.11 ± 1.645 * 0.04598

CI = -0.11 ± 0.07538

CI = (-0.18538, -0.03462)

The 90% confidence interval for the change in Obama's approval rating from October 2012 to October 2014 is (-0.18538, -0.03462).

Since the interval includes negative values, we can conclude that his approval rating decreased from October 2012 to October 2014.