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The city of Arequipa Peru takes a survey to gauge interest in a new rail system to Lima. 5499 of the respondents express support, 3463 of respondents express disapproval. What is the 95% confidence interval for support for the new service? Round to three decimal places.Upper bound Lower bound

Question

The city of Arequipa Peru takes a survey to gauge interest in a new rail system to Lima. 5499 of the respondents express support, 3463 of respondents express disapproval. What is the 95% confidence interval for support for the new service? Round to three decimal places.Upper bound Lower bound

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Solution

To calculate the 95% confidence interval for the proportion of support for the new rail system, follow these steps:

  1. Determine the sample proportion (p-hat): p^=Number of supportersTotal number of respondents=54995499+3463=54998962≈0.613 \hat{p} = \frac{\text{Number of supporters}}{\text{Total number of respondents}} = \frac{5499}{5499 + 3463} = \frac{5499}{8962} \approx 0.613

  2. Calculate the standard error (SE) of the proportion: SE=p^(1−p^)n=0.613×(1−0.613)8962=0.613×0.3878962≈0.2378962≈0.0000264≈0.005 SE = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} = \sqrt{\frac{0.613 \times (1 - 0.613)}{8962}} = \sqrt{\frac{0.613 \times 0.387}{8962}} \approx \sqrt{\frac{0.237}{8962}} \approx \sqrt{0.0000264} \approx 0.005

  3. Determine the z-score for a 95% confidence interval: The z-score for a 95% confidence interval is approximately 1.96.

  4. Calculate the margin of error (ME): ME=z×SE=1.96×0.005≈0.010 ME = z \times SE = 1.96 \times 0.005 \approx 0.010

  5. Determine the confidence interval: Lower bound=p^−ME=0.613−0.010=0.603 \text{Lower bound} = \hat{p} - ME = 0.613 - 0.010 = 0.603 Upper bound=p^+ME=0.613+0.010=0.623 \text{Upper bound} = \hat{p} + ME = 0.613 + 0.010 = 0.623

Therefore, the 95% confidence interval for support for the new rail system is approximately:

Lower bound=0.603 \text{Lower bound} = 0.603 Upper bound=0.623 \text{Upper bound} = 0.623

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