library(readr)
NutritionStudy <- read_csv("NutritionStudy.csv")
FishGills3 <- read_csv("FishGills3.csv")Homework 8
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
\(H_0\):\(p_1\) = \(p_2\) = \(p_3\) = 1/2
\(H_a\): at least on \(p_i\) \(\neq\) 1/2
#observed counts
observed <- c(244, 192)
#null values
theoritical_prop <- rep(1/2, 2)
#expected values
expected_values <- theoritical_prop*sum(observed)
expected_values[1] 218 218#values > 5, can perform chi sq test
chisq.test(observed)
Chi-squared test for given probabilities
data: observed
X-squared = 6.2018, df = 1, p-value = 0.01276Conclusion: Since the pvalue is 0.01276, we reject the null: we can conclude that the two options are equally likely
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 vitamin use and the gender of the participants
\(H_a\) : There is an association between vitamin use and the gender of the participants
#reate observed counts
observed_dataset<- table(NutritionStudy$VitaminUse, NutritionStudy$Sex)
observed_dataset
Female Male
No 87 24
Occasional 77 5
Regular 109 13#run anova test
chisq.test(observed_dataset)
Pearson's Chi-squared test
data: observed_dataset
X-squared = 11.071, df = 2, p-value = 0.003944Conclusion: The pvalue is 0.004, so reject the null. There is sufficient evidence that there is an association between vitamin use and the gender of the participants
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.
\(H_0\): \(\mu_A\) = \(\mu_B\) = \(\mu_C\)
\(H_a\): not all \(\mu_i\) are equal
#Anova test results
anova_result <- aov(GillRate ~ Calcium, data = FishGills3)
anova_resultCall:
aov(formula = GillRate ~ Calcium, data = FishGills3)
Terms:
Calcium Residuals
Sum of Squares 2037.222 19064.333
Deg. of Freedom 2 87
Residual standard error: 14.80305
Estimated effects may be unbalancedsummary(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 ' ' 1Conclusion: the pvalue is 0.012, so reject the null. There is sufficient evidence that there are significant differences in gill rate among the different calcium levels.