knitr::opts_chunk$set(echo = TRUE)##Problem 1
\(H_0\): \(p_1 = p_2 = 0.5\)
\(H_a\): At least one \(p_i \neq 0.5\) Where,
\(p_1\)= proportion of people with the R allele
\(p_2\) = proportion of people with the X allele
# Observed counts
observed <- c(244, 192)
# Null values
theoretical_prop <- rep(0.5, 2)
expected_values <- theoretical_prop*sum(observed)
expected_values## [1] 218 218chisq.test(observed, p= theoretical_prop)##
## Chi-squared test for given probabilities
##
## data: observed
## X-squared = 6.2018, df = 1, p-value = 0.01276P-value = 0.01276
Conclusion:With a p-value of 0.01276, which is less than the typical significance level of 0.05, there is sufficient evidence to reject the null hypothesis.The allele proportions are not equally likely in the population.
\(H_0\): There is no association between VitaminUse and Gender.
\(H_a\): There is an association between VitaminUse and Gender.
df <- read.csv("C:/Users/dspd2/OneDrive/Data 101/week 8/NutritionStudy.csv")
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. :3.000observed_dataset<- table(df$VitaminUse, df$Sex)
observed_dataset##
## Female Male
## No 87 24
## Occasional 77 5
## Regular 109 13chisq.test(observed_dataset)##
## Pearson's Chi-squared test
##
## data: observed_dataset
## X-squared = 11.071, df = 2, p-value = 0.003944Conclusion: With a p-value of 0.003944, which is less than the typical significance level of 0.05, there is sufficient evidence to reject the null hypothesis. Therefore, we conclude that there is a significant association between vitamin use and gender.
\(H_0\): \(\mu_A\) = \(\mu_B\) = \(\mu_C\)
\(H_a\): not all \(\mu_i\) are equal Where,
\(\mu_A\)= mean gill rate in low calcium
\(\mu_B\) = mean gill rate in medium calcium
\(\mu_C\) = mean gill rate in high calcium
Gr<-read.csv("C:/Users/dspd2/OneDrive/Data 101/week 8/FishGills3.csv")
summary(Gr)## Calcium GillRate
## Length:90 Min. :33.00
## Class :character 1st Qu.:48.00
## Mode :character Median :62.50
## Mean :61.78
## 3rd Qu.:72.00
## Max. :98.00anova_result <- aov(GillRate ~ Calcium, data = Gr)
anova_result## Call:
## aov(formula = GillRate ~ Calcium, data = Gr)
##
## 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 very small (9.12e-07): indicating strong evidence against the null hypothesis. Overall, this test suggests that there are significant differences in the mean gill rate depending on the calcium level of the water.