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## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors Your hypotheses. P-value Conclusion
#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
Hypothesis
\(H_0\):\(p_1\) = \(p_2\) = 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# Perform chi-squared goodness-of-fit test
chisq.test(observed)##
## Chi-squared test for given probabilities
##
## data: observed
## X-squared = 6.2018, df = 1, p-value = 0.01276Conclusion: The p-value is less than 0.05. We can conclude the result is statistically significant and the sample provides enough evidence to show that the two options are not 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)
setwd("~/Desktop/Data101 csv")
df<- 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.summary(df)## ID Age Smoke Quetelet
## Min. : 1.0 Min. :19.00 Length:315 Min. :16.33
## 1st Qu.: 79.5 1st Qu.:39.00 Class :character 1st Qu.:21.80
## Median :158.0 Median :48.00 Mode :character Median :24.74
## Mean :158.0 Mean :50.15 Mean :26.16
## 3rd Qu.:236.5 3rd Qu.:62.50 3rd Qu.:28.85
## Max. :315.0 Max. :83.00 Max. :50.40
## Vitamin Calories Fat Fiber
## Min. :1.000 Min. : 445.2 Min. : 14.40 Min. : 3.10
## 1st Qu.:1.000 1st Qu.:1338.0 1st Qu.: 53.95 1st Qu.: 9.15
## Median :2.000 Median :1666.8 Median : 72.90 Median :12.10
## Mean :1.965 Mean :1796.7 Mean : 77.03 Mean :12.79
## 3rd Qu.:3.000 3rd Qu.:2100.4 3rd Qu.: 95.25 3rd Qu.:15.60
## Max. :3.000 Max. :6662.2 Max. :235.90 Max. :36.80
## Alcohol Cholesterol BetaDiet RetinolDiet
## Min. : 0.000 Min. : 37.7 Min. : 214 Min. : 30.0
## 1st Qu.: 0.000 1st Qu.:155.0 1st Qu.:1116 1st Qu.: 480.0
## Median : 0.300 Median :206.3 Median :1802 Median : 707.0
## Mean : 3.279 Mean :242.5 Mean :2186 Mean : 832.7
## 3rd Qu.: 3.200 3rd Qu.:308.9 3rd Qu.:2836 3rd Qu.:1037.0
## Max. :203.000 Max. :900.7 Max. :9642 Max. :6901.0
## BetaPlasma RetinolPlasma Sex VitaminUse
## Min. : 0.0 Min. : 179.0 Length:315 Length:315
## 1st Qu.: 90.0 1st Qu.: 466.0 Class :character Class :character
## Median : 140.0 Median : 566.0 Mode :character Mode :character
## Mean : 189.9 Mean : 602.8
## 3rd Qu.: 230.0 3rd Qu.: 716.0
## Max. :1415.0 Max. :1727.0
## PriorSmoke
## Min. :1.000
## 1st Qu.:1.000
## Median :2.000
## Mean :1.638
## 3rd Qu.:2.000
## Max. :3.000Hypothesis
\(H_0\) : Gender is not associated with vitamin use \(H_a\) : Gender is associated with vitamin use
observed_dataset<- table(df$Sex, df$VitaminUse)
observed_dataset##
## No Occasional Regular
## Female 87 77 109
## Male 24 5 13chisq.test(observed_dataset)##
## Pearson's Chi-squared test
##
## data: observed_dataset
## X-squared = 11.071, df = 2, p-value = 0.003944Conclusion: The p-value of 0.003944 is less than the typical significance level of 0.05, so 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:
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
\(\mu_A\) = 0.71 mg/L (low) \(\mu_B\) = 5.24 mg/L (medium) \(\mu_C\) = 18.24 mg/L (high)
\(H_0\): \(\mu_A\) = \(\mu_B\) = \(\mu_C\)
\(H_a\): not all \(\mu_i\) are equal
setwd("~/Desktop/Data101 csv")
df<- 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.head(df)## # A tibble: 6 × 2
## Calcium GillRate
## <chr> <dbl>
## 1 Low 55
## 2 Low 63
## 3 Low 78
## 4 Low 85
## 5 Low 65
## 6 Low 98# Perform ANOVA
anova_result <- aov(GillRate ~ Calcium, data = df)
anova_result## Call:
## aov(formula = GillRate ~ Calcium, data = df)
##
## 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 p-value is small (0.0121): indicating strong evidence against the null hypothesis. Overall, this test suggests that there are significant differences in gill rates among the different calcium levels of the water.