Loading Datasets

library(tidyverse)
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fish_data <- read.csv("FishGills3.csv")
nutrition_data <- read.csv("NutritionStudy.csv")

Problem 1

ACTN3 is a gene that encodes alpha-actinin-3, a protein in fast-twitch muscle fibers, important for activities like sprinting and weightlifting. The gene has two main alleles: R (functional) and X (non-functional). The R allele is linked to better performance in strength, speed, and power sports, while the X allele is associated with endurance due to a greater reliance on slow-twitch fibers. However, athletic performance is influenced by various factors, including training, environment, and other genes, making the ACTN3 genotype just one contributing factor.

A study examines the ACTN3 genetic alleles R and X, also associated with fast-twitch muscles. Of the 436 people in this sample, 244 were classified as R, and 192 were classified as X. Does the sample provide evidence that the two options are not equally likely? Conduct the test using a chi-square goodness-of-fit test.

\(H_0\): Alleles are equally likely \(H_a\): Alleles are not equally likely

observed <- c(244, 192)

theoritical_prop <- rep(1/2, 2)

expected_values <- theoritical_prop*sum(observed)
expected_values
## [1] 218 218
chisq.test(observed)
## 
##  Chi-squared test for given probabilities
## 
## data:  observed
## X-squared = 6.2018, df = 1, p-value = 0.01276

With the p-value of 0.01276, we must reject the null hypothesis. There’s indication of significant evidence that the allele frequency aren’t equal.

Problem 2

Who Is More Likely to Take Vitamins: Males or Females? The dataset NutritionStudy contains, among other things, information about vitamin use and the gender of the participants. Is there a significant association between these two variables? Use the variables VitaminUse and Gender to conduct a chi-square analysis and give the results. (Test for Association)

\(H_0\): There is no association between Gender and Vitamin use \(H_a\): There is an association between Gender and Vitamin use

observed_dataset <- table(nutrition_data$VitaminUse, nutrition_data$Sex)
observed_dataset
##             
##              Female Male
##   No             87   24
##   Occasional     77    5
##   Regular       109   13
chisq.test(observed_dataset)
## 
##  Pearson's Chi-squared test
## 
## data:  observed_dataset
## X-squared = 11.071, df = 2, p-value = 0.003944

With the p-value of 0.003944, its indicating that there is significant association between gender and vitamin use, therefore we reject the null hypothesis.

Problem 3

Most fish use gills for respiration in water, and researchers can observe how fast a fish’s gill cover beats to study ventilation, much like we might observe a person’s breathing rate. Professor Brad Baldwin is interested in how water chemistry might affect gill beat rates. In one experiment, he randomly assigned fish to tanks with different calcium levels. One tank was low in calcium (0.71 mg/L), the second tank had a medium amount (5.24 mg/L), and the third tank had water with a high calcium level (18.24 mg/L). His research team counted gill rates (beats per minute) for samples of 30 fish in each tank. The results are stored in FishGills3. Perform ANOVA test to see if the mean gill rate differs depending on the calcium level of the water.

\(H_0\): The mean gill rate is the same in every calcium level \(H_a\): One mean gill rate differs amongst the calcium levels

anova_results <- aov(GillRate ~ Calcium, data = fish_data)
anova_results
## Call:
##    aov(formula = GillRate ~ Calcium, data = fish_data)
## 
## 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_results)
##             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

With the p-value of 0.0121, which is below from 0.05, therefore we must reject the null hypothesis. There is indication that the mean gill rate does indeed differ, depending on the calcium levels in the water.