樣本平均數機率計算

場景：

• 已知母體平均數（Population Mean），樣本數（Sample Size）。
• 已知母體平均數某區間（Sample Mean Range）。
• 求算不同樣本數，在母體平均數該區間之機率。

分析：

類似問題：

• 高爾夫杆數分析

GNU R：

# 已知母體平均數，樣本數，求算樣本標準差 SampleStdDeviation <- function(populationStdDeviation, sampleSize) { myStdDeviation <- populationStdDeviation / sqrt(sampleSize) myStdDeviation } # 已知母體平均數，樣本標準差，求算樣本在某母體平均數百分比區間之機率 SampleMeanPercentageRangeProbability <- function(populationMean, sampleStdDeviation, samplePercentageRange) { sampleValueRange <- populationMean * samplePercentageRange z2 <- sampleValueRange / sampleStdDeviation z1 <- -1 * z2 p2 <- pnorm(z2) p1 <- pnorm(z1) myProbability <- p2 - p1 myProbability } # 已知母體平均數，樣本標準差，求算樣本在某母體平均數值區間之機率 SampleMeanValueRangeProbability <- function(sampleStdDeviation, sampleValueRange) { z2 <- sampleValueRange / sampleStdDeviation z1 <- -1 * z2 p2 <- pnorm(z2) p1 <- pnorm(z1) myProbability <- p2 - p1 myProbability } ############################################################################################################## populationMean <- 4260 populationStdDeviation <- 900 sampleValueRange <- 100 seedSet <- seq(from=50, to=500, by=50) resultSet <- sampleSize for (mySampleSize in seedSet) { myStdDeviation <- SampleStdDeviation(populationStdDeviation, mySampleSize) print(sprintf("E(x)=%d σ=%.4f", populationMean, myStdDeviation)) myProbability <- SampleMeanValueRangeProbability(myStdDeviation, sampleValueRange) print(sprintf("P(%d)=%.4f σ=%.4f", sampleValueRange, myProbability, myStdDeviation)) pos <- mySampleSize / 50 resultSet[pos] <- myProbability } plot(main=sprintf("平均數=%d 標準差=%d 區間=%d", populationMean, populationStdDeviation, sampleValueRange), xlab="樣本數", ylab="機率", seedSet, resultSet)