An article describes surveys of representative samples of rural and urban adult Americans. Of 523 urban
adults, 195 reported that they use the Internet almost constantly. In the sample of 281 rural adults, 67
reported that they use the Internet almost constantly.
Construct a 99% confidence interval for the difference in the proportion who report using the Internet
almost constantly for urban and rural adult Americans. (Use Urban − Rural. Enter your answer using
interval notation. Round your numerical values to four decimal places.)
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To construct a 99% confidence interval for the difference in proportions, we can use the following formula:
Confidence Interval = (p̂₁ - p̂₂) ± Z * √[((p̂₁ * (1 - p̂₁))/n₁) + ((p̂₂ * (1 - p̂₂))/n₂)]
Where:
p̂₁ = proportion of urban adults who use the internet almost constantly
p̂₂ = proportion of rural adults who use the internet almost constantly
n₁ = sample size of urban adults
n₂ = sample size of rural adults
Z = Z-score for a 99% confidence level (2.58)
Calculating the proportions:
p̂₁ = 195/523 ≈ 0.373
p̂₂ = 67/281 ≈ 0.238
Calculating the confidence interval:
Confidence Interval = (0.373 - 0.238) ± 2.58 * √[((0.373 * (1 - 0.373))/523) + ((0.238 * (1 - 0.238))/281)]
Simplifying the formula:
Confidence Interval = 0.135 ± 2.58 * √[(0.245/523) + (0.146/281)]
Calculating the square roots and simplifying further:
Confidence Interval = 0.135 ± 2.58 * √[0.000295 + 0.000519]
Calculating the terms inside the square root:
Confidence Interval = 0.135 ± 2.58 * √0.000814
Calculating the square root:
Confidence Interval = 0.135 ± 2.58 * 0.028536
Calculating the multiplication:
Confidence Interval = 0.135 ± 0.073548
Calculating the interval:
Confidence Interval = (0.061452, 0.208548)
Therefore, the 99% confidence interval for the difference in the proportion who report using the Internet almost constantly for urban and rural adult Americans is (0.061452, 0.208548)
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