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.

Hypothesis:

H0: p1 = p2 = 1/2

H1: one p ≠ 1/2

observed_genes <- c(244, 192)

Theoretical_gene_proportions <- c(1/2, 2)

expected_genes <- Theoretical_gene_proportions*sum(observed_genes)
expected_genes
## [1] 218 872
chisq.test(observed_genes)
## 
##  Chi-squared test for given probabilities
## 
## data:  observed_genes
## X-squared = 6.2018, df = 1, p-value = 0.01276

P-Value: 0.0127

Conclusion: We reject the null hypothesis, R and X allele are not equally likely to occur.

  1. 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)

Hypothesis:

H0: Variables are independent.

H1: Variables are correlated.

NS <- read.csv("NutritionStudy.csv")

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

P-Value: 0.0039

Conclusion: We reject the null hypothesis, There is a correlation between gender and taking vitamins. Women are more likely than men to take vitamins.

  1. 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.

Hypothesis:

H0: µ Low = µ Medium = µ High

H1: µ is not equal in all three.

FG <- read.csv("FishGills3.csv")

rate <- aov(GillRate ~ Calcium, data = FG)
rate
## Call:
##    aov(formula = GillRate ~ Calcium, data = FG)
## 
## 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(rate)
##             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
TukeyHSD(rate)
##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = GillRate ~ Calcium, data = FG)
## 
## $Calcium
##                  diff        lwr        upr     p adj
## Low-High    10.333333   1.219540 19.4471264 0.0222533
## Medium-High  0.500000  -8.613793  9.6137931 0.9906108
## Medium-Low  -9.833333 -18.947126 -0.7195402 0.0313247

P-Value: 0.0121

Conclusion: We reject the null hypothesis, the mean gill rates were not the same and varied depending on calcium levels.