Question

the table shows the results of a survey in which separate samples of 400 adults each from the east, south, midwest, and west were asked if traffic congestion is a serious problem in their community. complete parts (a) and (b).
bad traffic congestion?
adults who say that traffic congestion is a serious problem
east 36%
south 31%
midwest 25%
west 53%
(a) construct a 95% confidence interval for the proportion of adults from the east who say traffic congestion is a serious problem.
the 95% confidence interval for the proportion of adults from the east who say traffic congestion is a serious problem is (.313, .407).
(round to three decimal places as needed.)
(b) construct a 95% confidence interval for the proportion of adults from the west who say traffic congestion is a serious problem.
the 95% confidence interval for the proportion of adults from the west who say traffic congestion is a serious problem is (□,□).
(round to three decimal places as needed.)

Explanation:

Step1: Identify key values

For West: $\hat{p}=0.53$, $n=400$, $z^*=1.96$ (95% confidence)

Step2: Calculate standard error

$\text{SE}=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\sqrt{\frac{0.53(1-0.53)}{400}}=\sqrt{\frac{0.53\times0.47}{400}}=\sqrt{\frac{0.2491}{400}}\approx0.02496$

Step3: Compute margin of error

$E=z^*\times\text{SE}=1.96\times0.02496\approx0.0489$

Step4: Find confidence interval

Lower bound: $0.53 - 0.0489 = 0.481$
Upper bound: $0.53 + 0.0489 = 0.579$

Answer:

The 95% confidence interval for the proportion of adults from the West who say traffic congestion is a serious problem is (0.481, 0.579)