家庭年收入誤差界限計算

場景：

• • 根據過去統計，家庭年平均收入為 $119155 元，標準差為 $30000 元。

  • 隨機調查 80 個家庭年收入。

問題：

• • 求算信心區間為 90% 下之誤差界限。（誤差界限=5517.0068 [113637.9932, 124672.0068]）

  • 求算信心區間為 95% 下之誤差界限。（誤差界限=6573.9191 [112581.0809, 125728.9191]）

  • 求算信心區間為 99% 下之誤差界限。（誤差界限=8639.5941 [110515.4059, 127794.5941]）

GNU R：

# 已知母體標準差，求誤差界限 # 樣本標準差 SampleStdDeviation <- function(populationStdDeviation, sampleSize) { sampleStdDeviation <- (populationStdDeviation) / (sampleSize - 1) sampleStdDeviation } # 樣本誤差 SampleStdError <- function(sampleStdDeviation) { sampleStdError <- sampleStdDeviation sampleStdError } # 從機率反求 z 值 z <- function(probability) { myZ <- qnorm(probability) myZ } # 誤差界限 MarginError <- function(populationStdDeviation, sampleSize, confidenceInterval) { myZ <- z(confidenceInterval + (1-confidenceInterval)/2) marginError <- myZ * (populationStdDeviation / sqrt(sampleSize)) marginError } PrintOut <- function(sampleSize, sampleMean, sampleStdError, confidenceInterval, marginError) { print(sprintf("樣本數=%4d 樣本平均數=%8.4f 樣本標準誤差=%.4f 信任區間=%.4f 誤差界限=%.4f [%8.4f, %8.4f]", sampleSize, sampleMean, sampleStdError, confidenceInterval, marginError, sampleMean-marginError, sampleMean+marginError)) } ############################################################################################################# # 已知 populationMean <- 119155 populationStdDeviation <- 30000 sampleSize <- 80 confidenceInterval <- c(0.9, 0.95, 0.99) # 求算 sampleMean <- populationMean sampleStdDeviation <- SampleStdDeviation(populationStdDeviation, sampleSize) sampleStdError <- SampleStdError(sampleStdDeviation) for (myConfidenceInterval in confidenceInterval) { marginError <- MarginError(populationStdDeviation, sampleSize, myConfidenceInterval) PrintOut(sampleSize, sampleMean, sampleStdError, myConfidenceInterval, marginError) } # 樣本數= 80 樣本平均數=119155.0000 樣本標準誤差=379.7468 信任區間=0.9000 誤差界限=5517.0068 [113637.9932, 124672.0068] # 樣本數= 80 樣本平均數=119155.0000 樣本標準誤差=379.7468 信任區間=0.9500 誤差界限=6573.9191 [112581.0809, 125728.9191] # 樣本數= 80 樣本平均數=119155.0000 樣本標準誤差=379.7468 信任區間=0.9900 誤差界限=8639.5941 [110515.4059, 127794.5941]