library(tidyverse)
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Nutrition_study <- read.csv("C:/Users/nika/Downloads/R/csv/NutritionStudy.csv")
Fish_gills <- read.csv("C:/Users/nika/Downloads/R/csv/FishGills3.csv")

Problem 1

observed_frequencies <- c(244, 192)
expected_frequencies <- c(218, 218)

\(H_0\):\(p_1\) = \(p_2\) = 1/2 \(H_a\): at least on \(p_i\) \(\neq\) 1/2

chisq.test(observed_frequencies)
## 
##  Chi-squared test for given probabilities
## 
## data:  observed_frequencies
## X-squared = 6.2018, df = 1, p-value = 0.01276

Based on the p value obtained, we reject the idea that the outcomes are equally likely and conclude that there are differences in the probabilities of X and R classification.

Problem 2

observed_dataset<- table(Nutrition_study$Vitamin, Nutrition_study$Sex)
observed_dataset
##    
##     Female Male
##   1    109   13
##   2     77    5
##   3     87   24

\(H_0\): There is no association between gender and vitamin use. \(H_a\): There is an association between gender and vitamin use.

chisq.test(observed_dataset)
## 
##  Pearson's Chi-squared test
## 
## data:  observed_dataset
## X-squared = 11.071, df = 2, p-value = 0.003944

With a p-value of 0.003944, which is less than the typical significance level of 0.05, there is sufficient evidence to reject the null hypothesis. Therefore, we conclude that there is a significant association between gender and vitamin use.

Problem 3

\(H_0\): \(\mu_A\) = \(\mu_B\) = \(\mu_C\) \(H_a\): not all \(\mu_i\) are equal

anova_result <- aov(GillRate ~ Calcium, data = Fish_gills)
anova_result
## Call:
##    aov(formula = GillRate ~ Calcium, data = Fish_gills)
## 
## Terms:
##                   Calcium Residuals
## Sum of Squares   2037.222 19064.333
## Deg. of Freedom         2        87
## 
## Residual standard error: 14.80305
## Estimated effects may be unbalanced
summary(anova_result)
##             Df Sum Sq Mean Sq F value Pr(>F)  
## Calcium      2   2037  1018.6   4.648 0.0121 *
## Residuals   87  19064   219.1                 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The p-value is smaller than the typical significance level of 0.05(0.0121): indicating strong evidence against the null hypothesis. Overall, this test suggests that there are significant differences in mean gill rates across the different calcium levels.