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
\(H_0\):\(p_1\) = \(p_2\) = 1/2
\(H_a\): at least on \(p_i\) \(\neq\) 1/2
Where:
\(p_1\) = genetic alleles R
\(p_2\) = genetic alleles X
α = 0.05
observed <- c(244, 192)
theoritical_prop <- rep(1/2, 2)
expected_values <- theoritical_prop*sum(observed)
expected_values## [1] 218 218Both values are greater than 5
chisq.test(observed)##
## Chi-squared test for given probabilities
##
## data: observed
## X-squared = 6.2018, df = 1, p-value = 0.01276As the p-value = 0.01276 which is less than 0.05 we can reject the null. The p-value is statistically significant at α = 0.05. We have enough evidence at a 0.05 significance level that the two options are not equally likely.
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)
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## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorssetwd("C:/Users/njnav/OneDrive/Data 101/Week 8/Import")
Nutrition <- read_csv("NutritionStudy.csv")## Rows: 315 Columns: 17
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (3): Smoke, Sex, VitaminUse
## dbl (14): ID, Age, Quetelet, Vitamin, Calories, Fat, Fiber, Alcohol, Cholest...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.observed_dataset<- table(Nutrition$VitaminUse, Nutrition$Sex)
observed_dataset##
## Female Male
## No 87 24
## Occasional 77 5
## Regular 109 13Hypothesis
\(H_0\) : Use of vitamins is not associated with gender
\(H_a\) : Use of vitamins is associated with gender
α = 0.05
chisq.test(observed_dataset)##
## Pearson's Chi-squared test
##
## data: observed_dataset
## X-squared = 11.071, df = 2, p-value = 0.003944As the p-value = 0.003944 which is less than 0.05 we can reject the null. The p-value is statistically significant at α = 0.05. We have enough evidence at a 0.05 significance level that there is a significant association between use of vitamins and gender.
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.
setwd("C:/Users/njnav/OneDrive/Data 101/Week 8/Import")
Fish <- read_csv("FishGills3.csv")## Rows: 90 Columns: 2
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (1): Calcium
## dbl (1): GillRate
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.Hypothesis
\(H_0\): \(\mu_A\) = \(\mu_B\) = \(\mu_C\)
\(H_a\): not all \(\mu_i\) are equal
Where:
\(\mu_A\) = mean gill rate in low calcium level water
\(\mu_B\) = mean gill rate in medium calcium level water
\(\mu_C\) = mean gill rate in high calcium level water
α = 0.05
anova_result <- aov(GillRate ~ Calcium, data = Fish)
anova_result## Call:
## aov(formula = GillRate ~ Calcium, data = Fish)
##
## 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 ' ' 1As the p-value = 0.0121 which is less than 0.05 we can reject the null. The p-value is statistically significant at α = 0.05. We have enough evidence at a 0.05 significance level that the mean gill rate differs depending on the calcium level of the water.