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# section 3.3 Statistical Methods for Evaluation
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# section 3.3.1 Hypothesis Testing
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# generate random observations from the two populations
x <- rnorm(10, mean=100, sd=5) # normal distribution centered at 100
y <- rnorm(20, mean=105, sd=5) # normal distribution centered at 105
# Student's t-test
t.test(x, y, var.equal=TRUE) # run the Student's t-test
##
## Two Sample t-test
##
## data: x and y
## t = -3.0312, df = 28, p-value = 0.0052
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -11.90850 -2.30399
## sample estimates:
## mean of x mean of y
## 99.8523 106.9585
# obtain t value for a two-sided test at a 0.05 significance level
qt(p=0.05/2, df=28, lower.tail= FALSE)
## [1] 2.048407
# Welch's t-test
t.test(x, y, var.equal=FALSE) # run the Welch's t-test
##
## Welch Two Sample t-test
##
## data: x and y
## t = -2.5236, df = 11.911, p-value = 0.02686
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -13.246741 -0.965751
## sample estimates:
## mean of x mean of y
## 99.8523 106.9585
# Wilcoxon Rank-Sum Test
wilcox.test(x, y, conf.int = TRUE)
##
## Wilcoxon rank sum test
##
## data: x and y
## W = 43, p-value = 0.01108
## alternative hypothesis: true location shift is not equal to 0
## 95 percent confidence interval:
## -13.318635 -1.765371
## sample estimates:
## difference in location
## -7.222526
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# section 3.3.6 ANOVA
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offers <- sample(c("offer1", "offer2", "nopromo"), size=500, replace=T)
# Simulated 500 observations of purchase sizes on the 3 offer options
purchasesize <- ifelse(offers=="offer1", rnorm(500, mean=80, sd=30),
ifelse(offers=="offer2", rnorm(500, mean=85, sd=30),
rnorm(500, mean=40, sd=30)))
# create a data frame of offer option and purchase size
offertest <- data.frame(offer=as.factor(offers),
purchase_amt=purchasesize)
# display a summary of offertest where offer="offer1"
summary(offertest[offertest$offer=="offer1",])
## offer purchase_amt
## nopromo: 0 Min. : 4.344
## offer1 :171 1st Qu.: 54.322
## offer2 : 0 Median : 73.341
## Mean : 76.625
## 3rd Qu.: 97.037
## Max. :166.014
# display a summary of offertest where offer="offer2"
summary(offertest[offertest$offer=="offer2",])
## offer purchase_amt
## nopromo: 0 Min. : 14.97
## offer1 : 0 1st Qu.: 65.81
## offer2 :169 Median : 88.32
## Mean : 87.07
## 3rd Qu.:107.46
## Max. :155.58
# display a summary of offertest where offer="nopromo"
summary(offertest[offertest$offer=="nopromo",])
## offer purchase_amt
## nopromo:160 Min. :-45.66
## offer1 : 0 1st Qu.: 12.04
## offer2 : 0 Median : 36.15
## Mean : 36.33
## 3rd Qu.: 58.93
## Max. :105.14
# fit ANOVA test
model <- aov(purchase_amt ~ offers, data=offertest)
summary(model)
## Df Sum Sq Mean Sq F value Pr(>F)
## offers 2 234375 117187 129.3 <2e-16 ***
## Residuals 497 450373 906
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Tukey's Honest Significant Difference (HSD) on all
# pair-wise tests for difference of means
TukeyHSD(model)
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = purchase_amt ~ offers, data = offertest)
##
## $offers
## diff lwr upr p adj
## offer1-nopromo 40.29274 32.509310 48.07618 0.0000000
## offer2-nopromo 50.73950 42.933841 58.54517 0.0000000
## offer2-offer1 10.44676 2.771144 18.12238 0.0041754