Файлы решений

R scripts \ R скрипты

Normality check \ Проверки нормальности

#1

shapiro.test(X) # Тест Шапиро

# Pvalue<alfa=0.05 гипотезу о нормальности отвергаем

#2

library(ggpubr)
ggdensity(X,main = "Density plot of X", xlab = "X")
glot(X)gqqp

#3

library(nortest) # Библиотека для pearson.test и lillie.test
pearson.test(X$V1) # Пирсон
lillie.test(X$V1)

#4

library("sm") # Для sm.density
sm.density(X$V1, model = "Normal") # Попадаем в коридор?

Dice&Events space

Pearson Spearman Kendall Correlation

Clean the data \ Очистка данных

#MAC:
x <- as.numeric(read.delim(pipe("pbpaste"), header = F, sep = ","))
#PC:
x<- as.numeric(read.delim("clipboard", header = F, sep = ","))

#delete NAN and string values
y <- na.omit(x)

Presentations \ Презентации

Hypothesis Tests

1. Mean test (normal dist., known variance)

R: z.test() (from BSDA package)

Excel: Z.TEST(array, μ₀, σ) Data Analysis: One-sample z-test

2. Mean test (normal dist., unknown variance)

R: t.test(x, mu = μ₀)

Excel: T.TEST(array, μ₀, tails, type=1) Data Analysis: One-sample t-test

3. Equality of two means (independent samples, known variances)

R: z.test(x, y, sigma.x=σ₁, sigma.y=σ₂)

Excel: Two-sample z-test

4. Equality of two means (independent samples, unknown but equal variances)

R: t.test(x, y, var.equal=TRUE)

Example: male<-c(26200, 24700, 28400, 21700) female<-c(22600, 23600, 29300, 22300)
t.test(male, female, "greater", paired=TRUE)

Excel: T.TEST(array1, array2, tails, type=2) Data Analysis: Student's t-test (equal variances)

5. Equality of two means (independent samples, unknown & unequal variances)

t.test(x, y, var.equal=FALSE)

Example: Test the null hypothesis H0: E(X)=E(Y) without the assumption of equality of variances at significance level α=0.02 against the alternative H1: E(X)>E(Y)

t.test(X,Y, alternative="greater", var.equal=FALSE, conf.level=0.98)

Excel: T.TEST(array1, array2, tails, type=3) Data Analysis: Welch’s t-test (unequal variances)

6. Equality of means (paired data)

t.test(x, y, paired=TRUE)

Excel: T.TEST(array1, array2, tails, type=1) Data Analysis: Paired t-test

7. Equality of variances (two samples)

var.test(x, y)

Example: Test the null hypothesis H0: Var(X)=Var(Y) at significance level α=0.05 against the alternative H1: Var(X)≠Var(Y). var.test(x,y, alternative='two.sided' )

Excel: F.TEST(array1, array2) Data Analysis: F-test (Fisher’s test)

8. Equality of proportions

prop.test(c(x_succ, y_succ), c(n₁, n₂), correct = FALSE)

Example: test<-prop.test(c(191,145), c(381,166), alternative = "two.sided", correct = FALSE)
-sqrt(test$statistic) # Z-statistic (where z² equals the chi-squared statistic for proportions) is negative because prop1 is less than prop2: 191/381 < 145/166

Excel: ---

9. Significance of correlation

cor.test(x, y, method="pearson")

Example: Test the hypothesis for insignificance of correlation coefficient ρ (i.e. H0: ρ=0 against the alternative H1: ρ≠0) cor.test(x, y, alternative = "two.sided")

Excel: PEARSON(array1, array2) + T.DIST for p-value Data Analysis: Pearson correlation test

Pareto optimum \ Оптимум по Парето

library(rPref)
df <- data.frame(Object= c("A", "B", "C", "D", "E"),
Tech = c(5, 6, 3, 6, 4),Art = c(6, 1, 5, 5, 3))
pref <- high(Tech) * high(Art)
front <- psel(df, pref)
plot(df$Tech, df$Art,
type = "n",xlab = "Technique",ylab = "Artistry",
main = "Pareto-front",xlim = c(0, 7),ylim = c(0, 7))
points(front$Tech, front$Art, col = "red", pch = 19, cex = 1.5)
others <- df[!df$Object%in% front$Object, ]
points(others$Tech, others$Art, col = "gray", pch = 1, cex = 1.5)
text(df$Tech, df$Art, labels = df$Object, pos = 4, cex = 0.9)
if (nrow(front) >= 2) {front_sorted <- front[order(front$Tech), ]
lines(front_sorted$Tech, front_sorted$Art,
col = "red", lty = 2, lwd = 2)}
legend("topright",legend = c("Pareto-opt", "Dominated"),
col = c("red", "gray"),pch = c(19, 1),bty = "n")

CI for E(x) \ Интервал для матожидания

install.packages("BSDA")
library(BSDA)

data<-c(18.2, 13.7, 15.9, 17.4, 21.8, 16.6, 12.3, 18.8, 16.2)
z.test(x = data,
sigma.x = 3.8, # стандартное отклонение
mu = 2.90, # выборочное среднее
conf.level = 0.9,
alternative = "two.sided")