Below are the solutions to these exercises on data display.

#################### # # # Exercise 1 # # # #################### n <- 100 p <- 0.3 #a dbinom(34, n, p)

## [1] 0.05788395

sum(dbinom(34:n, n, p))

## [1] 0.2207422

pbinom(34, n, p)

## [1] 0.8371417

#b sum(dbinom(30:60, n, p))

## [1] 0.5376603

#c qbinom(0.025,n,p)

## [1] 21

qbinom(0.975,n,p)

## [1] 39

#################### # # # Exercise 2 # # # #################### m <- 3 s <- 1 #a pnorm(2,m,s)

## [1] 0.1586553

pnorm(4,m,s) - pnorm(2,m,s)

## [1] 0.6826895

#b qnorm(0.025,m,s)

## [1] 1.040036

qnorm(0.975,m,s)

## [1] 4.959964

qnorm(0.5,m,s)

## [1] 3

#################### # # # Exercise 3 # # # #################### df <- 8 #a pt(1,df)

## [1] 0.8267032

1-pt(2,df)

## [1] 0.04025812

pt(1,df)-pt(-1,df)

## [1] 0.6534065

#b qt(0.025,df)

## [1] -2.306004

qt(0.5,df)

## [1] 0

1-qt(0.075,df)

## [1] 2.592221

#################### # # # Exercise 4 # # # #################### df <- 5 #a pchisq(2,df)

## [1] 0.150855

1-pchisq(4,df)

## [1] 0.549416

# OR pchisq(4,df,lower.tail = FALSE)

## [1] 0.549416

pchisq(6,df)-pchisq(4,df)

## [1] 0.243197

#b qchisq(0.025, df, lower.tail=TRUE)

## [1] 0.8312116

qchisq(0.5, df, lower.tail=TRUE)

## [1] 4.35146

qchisq(0.075, df, lower.tail=FALSE)

## [1] 10.00831

#################### # # # Exercise 5 # # # #################### df_1 <- 6 df_2 <- 3 pf(2, df_1, df_2)

## [1] 0.6958948

1 - pf(3, df_1, df_2)

## [1] 0.1977977

pf(4, df_1, df_2) - pf(1, df_1, df_2)

## [1] 0.4039858

qf(0.025,df_1, df_2)

## [1] 0.1515427

qf(0.975,df_1, df_2)

## [1] 14.73472

#################### # # # Exercise 6 # # # #################### data <- data.frame(case = factor(rep(c("A","B","C"), each=100)), gen = c(rbinom(100, 20, 0.3), rbinom(100, 20, 0.5), rbinom(100, 20, 0.7))) ggplot(data, aes(x=gen, fill=case)) + geom_density(alpha=.3)

#################### # # # Exercise 7 # # # #################### data <- data.frame(case = factor(rep(c("A","B","C"), each=100)), gen = c(rnorm(100, 0, 1), rnorm(100, 0, 3), rnorm(100, 0, 7))) ggplot(data, aes(x=gen, fill=case)) + geom_density(alpha=.3)

#################### # # # Exercise 8 # # # #################### data <- data.frame(case = factor(rep(c("A","B","C"), each = 100)), gen = c(rt(100, 5), rt(100, 10), rt(100, 25))) ggplot(data, aes(x=gen, fill=case)) + geom_density(alpha=.3)

#Notice the variance, which decreases as the degrees of freedom increase #################### # # # Exercise 9 # # # #################### data <- data.frame(case = factor(rep(c("A","B","C"), each = 100)), gen = c(rchisq(100, 5), rchisq(100, 10), rchisq(100, 25))) ggplot(data, aes(x=gen, fill=case)) + geom_density(alpha=.3)

#Observe that the graphs change from heavily skew to the right into more bell-shaped. #################### # # # Exercise 10 # # # #################### data <- data.frame(case = factor(rep(c("A","B","C"), each = 100)), gen = c(rf(100, 3, 9), rf(100, 9, 3), rf(100,15, 15))) ggplot(data, aes(x=gen, fill=case)) + geom_density(alpha=.3)+xlim(0, 10)

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