公司董事長為女性機率計算
場景：
假定公司董事長平均有 28% 為女性。
假定隨機抽樣 240 家公司。
問題：
求算樣本標準差。（0.0181）
求算樣本分佈曲線。
求算樣本在平均值 +- 0.04 區間內之機率。（0.9731）
求算樣本在平均值 +- 0.02 區間內之機率。（0.7315）
GNU R：
# 樣本分佈平均值 SampleDistributionMean <- function(populationMean) { myMean <- populationMean myMean } # 已知母體平均數，樣本數，求算樣本標準差 SampleDistributionStdDeviation <- function(populationStdDeviation, sampleSize) { myStdDeviation <- populationStdDeviation / sqrt(sampleSize) myStdDeviation } # 樣本分佈標準誤差 SampleDistributionStdError <- function(sampleDistributionStdDeviation) { myStdError <- sampleDistributionStdDeviation myStdError } # 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 SampleDistributionPercentageRangeProbability <- function(populationMean, sampleStdDeviation, samplePercentageRange) { sampleValueRange <- populationMean * samplePercentageRange z2 <- sampleValueRange z1 <- -1 * z2 p2 <- pnorm(z2, sd=sampleStdDeviation) p1 <- pnorm(z1, sd=sampleStdDeviation) 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 SampleDistributionValueRangeProbability <- function(sampleStdDeviation, sampleValueRange) { z2 <- sampleValueRange z1 <- -1 * z2 p2 <- pnorm(z2, sd=sampleStdDeviation) p1 <- pnorm(z1, sd=sampleStdDeviation) 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 SampleDistributionProbability <- function(sampleMean, sampleStdDeviation, sampleValue) { z <- (sampleValue - sampleMean) / sampleStdDeviation myProbability <- pnorm(z) myProbability } PlotNormalDistributionChart <- function(meanValue, stdDeviation) { myChartTitle <- sprintf("平均值=%.2f 標準差=%.4f", meanValue, stdDeviation) curve(main=myChartTitle, exp((-1 * (x - meanValue)^2)/(2 * stdDeviation ^ 2))/(stdDeviation * sqrt(2 * pi)), from=meanValue - 3 * stdDeviation, to=meanValue + 3 * stdDeviation, n=100, xlab="值", ylab="機率密度") myChartTitle } ############################################################################################################## populationMean <- 0.28 sampleSize <- 240 sampleMean <- SampleDistributionMean(populationMean) sampleStdDeviation <- SampleDistributionStdDeviation(populationMean, sampleSize) print(sprintf("平均值=%.2f 標準差=%.4f", sampleMean, sampleStdDeviation)) sampleValueRange <- 0.04 myProbability <- SampleDistributionValueRangeProbability(sampleStdDeviation, sampleValueRange) print(sprintf("區間介於 %.2f 機率=%.4f", sampleValueRange, myProbability)) sampleValueRange <- 0.02 myProbability <- SampleDistributionValueRangeProbability(sampleStdDeviation, sampleValueRange) print(sprintf("區間介於 %.2f 機率=%.4f", sampleValueRange, myProbability)) PlotNormalDistributionChart(sampleMean, sampleStdDeviation)
