小企業網站維護費用誤差界限計算

場景：

• • 根據過去統計，小企業維護網站年平均花費為 $11500 元，標準差為 $4000 元。

  • 隨機調查 60 家小企業維護網站年平均花費。

問題：

• • 求算誤差界限。（1012.1210）

  • 若需要信心區間為 95% 下，求算小企業調查家數。（245）

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 <- 11500 populationStdDeviation <- 4000 sampleSize <- 60 confidenceInterval <- 0.95 # 求算 sampleMean <- populationMean sampleStdDeviation <- SampleStdDeviation(populationStdDeviation, sampleSize) sampleStdError <- SampleStdError(sampleStdDeviation) marginError <- MarginError(populationStdDeviation, sampleSize, confidenceInterval) PrintOut(sampleSize, sampleMean, sampleStdError, confidenceInterval, marginError) # 樣本數= 60 樣本平均數=11500.0000 樣本標準誤差=67.7966 信任區間=0.9500 誤差界限=1012.1210 [10487.8790, 12512.1210] # 已知 marginError <- 500 # 求算 sampleSize <- as.integer(((z(confidenceInterval + (1-confidenceInterval)/2) * populationStdDeviation) / marginError) ^ 2) sampleStdDeviation <- SampleStdDeviation(populationStdDeviation, sampleSize) sampleStdError <- SampleStdError(sampleStdDeviation) PrintOut(sampleSize, sampleMean, sampleStdError, confidenceInterval, marginError) # 樣本數= 245 樣本平均數=11500.0000 樣本標準誤差=16.3934 信任區間=0.9500 誤差界限=500.0000 [11000.0000, 12000.0000]