Below are the solutions to these exercises on inferential statistics.

#################### # # # Exercise 1 # # # #################### binom.test(5 ,30, mean(data$class),alternative = "two.sided")

## ## Exact binomial test ## ## data: 5 and 30 ## number of successes = 5, number of trials = 30, p-value = 0.03587 ## alternative hypothesis: true probability of success is not equal to 0.3489583 ## 95 percent confidence interval: ## 0.0564217 0.3472117 ## sample estimates: ## probability of success ## 0.1666667

#################### # # # Exercise 2 # # # #################### binom.test(c(5, 25), mean(data$class) ,alternative = "two.sided")

## ## Exact binomial test ## ## data: c(5, 25) ## number of successes = 5, number of trials = 30, p-value = ## 0.0003249 ## alternative hypothesis: true probability of success is not equal to 0.5 ## 95 percent confidence interval: ## 0.0564217 0.3472117 ## sample estimates: ## probability of success ## 0.1666667

#################### # # # Exercise 3 # # # #################### binom.test(5, 30, mean(data$class), alternative="less")

## ## Exact binomial test ## ## data: 5 and 30 ## number of successes = 5, number of trials = 30, p-value = 0.0239 ## alternative hypothesis: true probability of success is less than 0.3489583 ## 95 percent confidence interval: ## 0.0000000 0.3189712 ## sample estimates: ## probability of success ## 0.1666667

#OR pbinom(5, 30, mean(data$class))

## [1] 0.0238959

# We reject our null hypothesis #################### # # # Exercise 4 # # # #################### binom.test(5,30, mean(data$class), conf.level=0.99,alternative="less")

## ## Exact binomial test ## ## data: 5 and 30 ## number of successes = 5, number of trials = 30, p-value = 0.0239 ## alternative hypothesis: true probability of success is less than 0.3489583 ## 99 percent confidence interval: ## 0.0000000 0.3808047 ## sample estimates: ## probability of success ## 0.1666667

# we can't reject our null hypothesis #################### # # # Exercise 5 # # # #################### binom.test(2, 30, mean(data$class), conf.level=0.999,alternative="less")

## ## Exact binomial test ## ## data: 2 and 30 ## number of successes = 2, number of trials = 30, p-value = ## 0.0003637 ## alternative hypothesis: true probability of success is less than 0.3489583 ## 99.9 percent confidence interval: ## 0.0000000 0.3214435 ## sample estimates: ## probability of success ## 0.06666667

# We reject our null hypothesis #################### # # # Exercise 6 # # # #################### z <- 1.96 low <- mean(data$mass) - z*sd(data$mass)/sqrt(30) high <- mean(data$mass) + z*sd(data$mass)/sqrt(30) low;high

## [1] 29.17127

## [1] 34.81389

#################### # # # Exercise 7 # # # #################### z <- (29 - mean(data$mass))/(sd(data$mass)/sqrt(30)) #################### # # # Exercise 8 # # # #################### 2*pnorm(-abs(z),0,1) #Reject the null hypothesis

## [1] 0.03761903

#################### # # # Exercise 9 # # # #################### library(TeachingDemos) z.test(29,mu=mean(data$mass),sd=sd(data$mass)/sqrt(30), alternative = "two.sided", conf.level = 0.95)

## ## One Sample z-test ## ## data: 29 ## z = -2.079, n = 1.0000, Std. Dev. = 1.4394, Std. Dev. of the ## sample mean = 1.4394, p-value = 0.03762 ## alternative hypothesis: true mean is not equal to 31.99258 ## 95 percent confidence interval: ## 26.17874 31.82126 ## sample estimates: ## mean of 29 ## 29

#################### # # # Exercise 10 # # # #################### z.test(29,mu=mean(data$mass),sd=sd(data$mass)/sqrt(30), alternative = "less", conf.level = 0.99)

## ## One Sample z-test ## ## data: 29 ## z = -2.079, n = 1.0000, Std. Dev. = 1.4394, Std. Dev. of the ## sample mean = 1.4394, p-value = 0.01881 ## alternative hypothesis: true mean is less than 31.99258 ## 99 percent confidence interval: ## -Inf 32.34865 ## sample estimates: ## mean of 29 ## 29

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