require(mosaic)
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library(readr)
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#Read data set Games1 into your R session and name it "games1":
games1 <- read_csv("P:/SUNY-Brockport/SUNY Brockport_1/teaching/Staitstical Methods/Spring 2017/Chapter2/C2Games1.csv")
## Parsed with column specification:
## cols(
## studentID = col_integer(),
## Type = col_character(),
## Time = col_integer()
## )
View(games1)
#Create a boxplot to compare the times for the color and standard grames:
boxplot(Time~Type,data=games1,ylab="Seconds")
#Create two vectors of Time observations for the Standard and Color groups:
Standard <- games1[games1$Type=="Standard",]$Time
Color <- games1[games1$Type=="Color",]$Time
#Mean and standard deviations for the time to win the game for standard game:
mean(Standard)
## [1] 35.55
sd(Standard)
## [1] 3.394655
#Mean and standard deviations for the time to win the game for color game:
mean(Color)
## [1] 38.1
sd(Color)
## [1] 3.654845
#Sample size, 12th observation, and 12th residual for the Standard group:
length(Standard)
## [1] 20
Standard[12]
## [1] 37
Standard[12]-mean(Standard)
## [1] 1.45
#Sample size, 12th observation, and 12th residual for the Color group:
length(Color)
## [1] 20
Color[12]
## [1] 37
Color[12]-mean(Color)
## [1] -1.1
#Create a vector of residuals for all observations:
residuals <- c(Standard-mean(Standard),Color-mean(Color))
#Create a histogram of the residuals:
hist(residuals)
#Compute the ratio of the maximum of the sample standard deviations for the Standard and Color groups to
#the minimum of sample standard deviations for the Standard and Color groups:
max(c(sd(Standard),sd(Color)))/min(c(sd(Standard),sd(Color)))
## [1] 1.076647
###############
########t-test
#Perform a two-sample t-test to compare average winning time between the Standard and Color groups
#(assuming equal variances):
t.test(Time~Type,data=games1,var.equal = TRUE)
##
## Two Sample t-test
##
## data: Time by Type
## t = 2.2862, df = 38, p-value = 0.02791
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.2920254 4.8079746
## sample estimates:
## mean in group Color mean in group Standard
## 38.10 35.55
# In this chapter we are assuming equal variances. Using “var.equal = FALSE” will give the more common t-test with unequal variances.
###########################
#Regression##################
#Create a dummy variable where a value of 1 corresponds to the color distracter game:
D1 <- (games1$Type=="Color")*1
# this creates a new column 1 for Color and 0 for Standard
#Regress Time on the dummy variable D1 and name the regression object games.reg1:
games.reg1 <- lm(Time~D1,data=games1)
#Obtain the vector of residuals:
residuals <- games.reg1$res
#Estimated coefficients of the model of Time regressed on D1:
games.reg1$coef
## (Intercept) D1
## 35.55 2.55
beta1<-games.reg1$coef[2]
#The partial output from the following code contains the t-statistic and p-value for
#the hypothesis test that β1 0 versus the alternative β1 ≠0:
summary(games.reg1)
##
## Call:
## lm(formula = Time ~ D1, data = games1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.100 -2.550 0.175 2.038 7.900
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 35.5500 0.7887 45.074 <2e-16 ***
## D1 2.5500 1.1154 2.286 0.0279 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.527 on 38 degrees of freedom
## Multiple R-squared: 0.1209, Adjusted R-squared: 0.09778
## F-statistic: 5.227 on 1 and 38 DF, p-value: 0.02791
# The 95% confidence interval for β1 is computed by:
LCL<-beta1 - qt(.975,nrow(games1)-2)*1.1154
UCL<-beta1 + qt(.975,nrow(games1)-2)*1.1154
# 2.55 is the slope coefficient and 1.1154 is the corresponding standard deviation.
#Construct a histogram of the residuals and a residual versus order plot:
par(mfrow=c(1,2))
hist(residuals)
plot(games1$studentID,residuals,xlab="Order",ylab="Residuals")
lines(games1$studentID,residuals)
abline(1,0)
#Create scatterplot of Time versus Type of game with a superimposed regression line:
plot(D1,games1$Time)
abline(lm(Time~D1,data=games1)$coef)
####################
#ANOVA################
#Compute y.. , 1. y , and 2. y , respectively:
mean(games1$Time)
## [1] 36.825
mean(Standard)
## [1] 35.55
mean(Color)
## [1] 38.1
#Estimate the effect size for the color distracter game:
mean(Color)-mean(games1$Time)
## [1] 1.275
#Estimate the effect size for the standard game:
mean(Standard)-mean(games1$Time)
## [1] -1.275
#Construct the main effects plot of game type:
par(mfrow=c(1,1))
plot(c(0,1),c(mean(Color),mean(Standard)),xlim=c(-.1,1.1),ylim=c(35.2,38.5), ylab="Mean Time",pch=16,xlab="Type",main="Main Effects Plot")
lines(c(0,1),c(mean(Color),mean(Standard)))
text(c(0,1),c(35.2,35.2),c("Color","Standard"))
#Obtain the results (F-statistic and p-value) of an analysis of variance:
summary(aov(Time~Type,data=games1))
## Df Sum Sq Mean Sq F value Pr(>F)
## Type 1 65.0 65.02 5.227 0.0279 *
## Residuals 38 472.8 12.44
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
sqrt(5.227)
## [1] 2.286263
# Create a vector of residuals from the analysis of variance model:
residuals1 <- aov(Time~Type,data=games1)$res
#Construct a histogram of the residuals, a plot of the residuals versus type of game, and a plot of the residuals
#versus order (all in one graphics window):
par(mfrow=c(1,3))
hist(residuals1,xlab="Residuals")
#plot(games1$Type,residuals,ylab="Residuals")
plot(games1$studentID,residuals,xlab="Order",ylab="Residuals")
lines(games1$studentID,residuals1)
abline(h = 0)
