NS<-read.csv("NutritionStudy.csv", header = TRUE)
  1. Find a 99% confidence interval for the average number of grams of fiber per day that people eat. Interpret the results.
print("H₀:  μ = μ₀    H₁: μ ≠ μ₀")
## [1] "H₀:  μ = μ₀    H₁: μ ≠ μ₀"
### This is a one sample t-test because the only data we're interested in is how much fiber people are eating daily

# Calculate the confidence interval
confidence_interval <- t.test(NS$Fiber, conf.level = 0.99)$conf.int
summary(NS$Fiber)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    3.10    9.15   12.10   12.79   15.60   36.80
print("The average amount of fiber people eat each day is 12.79 grams")
## [1] "The average amount of fiber people eat each day is 12.79 grams"
  1. Construct 95% confidence interval for the effect of gender on smoking.
gend_smoke <- subset(NS, select = c(Sex, Smoke))

male_smokers <- sum(gend_smoke$Sex == "Male" & gend_smoke$Smoke == "Yes")
female_smokers <- sum(gend_smoke$Sex == "Female" & gend_smoke$Smoke == "Yes")

total_smokers <- male_smokers + female_smokers

percent_male_smokers <- (male_smokers / total_smokers) * 100
percent_female_smokers <- (female_smokers / total_smokers) * 100

pie(c(percent_male_smokers, percent_female_smokers),
    labels = c("Male", "Female"),
    main = "Percentage of Smokers by Gender",
    col = c("blue", "pink"))

b) Do the sample data provide evidence that the effect of gender on smoking is statistically significant? In other words, Is there evidence of a difference in the percentage of current smokers by gender?

cat("There is statistically significant data showing gender affects smoking")
## There is statistically significant data showing gender affects smoking
cat("Percent Male Smokers: 16.28%        Percent Female Smokers: 83.72%")
## Percent Male Smokers: 16.28%        Percent Female Smokers: 83.72%
  1. Does smoking decrease beta-carotene levels in the blood (betaPlasma)? Justify your answer with confidence interval and a p-value.
NS$Smoke <- as.factor(NS$Smoke)
t_test_result <- t.test(BetaPlasma ~ Smoke, data = NS)
ci <- t_test_result$conf.int
p_value <- t_test_result$p.value
cat("95% Confidence Interval for the difference in mean BetaPlasma levels between smokers and non-smokers:", ci[1], "to", ci[2], "\n")
## 95% Confidence Interval for the difference in mean BetaPlasma levels between smokers and non-smokers: 46.29873 to 112.5133
cat("P-value:", p_value, "\n")
## P-value: 5.174922e-06
cat("There is statistically significant evidence that smoking affects the beta-carotene levels in the blood.\n")
## There is statistically significant evidence that smoking affects the beta-carotene levels in the blood.
library(ggplot2)
ggplot(NS, aes(x = Smoke, y = BetaPlasma, color = Smoke)) + geom_point()

  1. Construct 90% confidence interval for the effect of sex on beta-carotene levels in the blood.
NS$Sex <- as.factor(NS$Sex)
t_test_sex <- t.test(NS$BetaPlasma ~ NS$Sex, conf.level = 0.90)
ci_sex_90 <- t_test_sex$conf.int
cat("90% Confidence Interval for difference in Beta Plasma levels in Males VS Females:", ci_sex_90[1], "to", ci_sex_90[2], "\n")
## 90% Confidence Interval for difference in Beta Plasma levels in Males VS Females: 8.26892 to 86.86661
  1. Does sex have any significant effect on beta-carotene levels in the blood?
t_test_result <- t.test(NS$BetaPlasma ~ NS$Sex)
p_value <- t_test_result$p.value
cat("P-value:", p_value, "\n")
## P-value: 0.04747518
cat("Sex has a statistically significant effect on Beta-Carotene levels in the blood. \n")
## Sex has a statistically significant effect on Beta-Carotene levels in the blood.
  1. Are the amount of alcohol consumption and the age of the people significantly related?
cor_test_result <- cor.test(NS$Age, NS$Alcohol)
cat("Correlation Coefficient:", cor_test_result$estimate, "\n")
## Correlation Coefficient: 0.05158327
cat("P-value:", cor_test_result$p.value, "\n")
## P-value: 0.361516
cat("There isn't a statistically significance relationship between age and alcohol consumption.\n")
## There isn't a statistically significance relationship between age and alcohol consumption.
library(ggplot2)
ggplot(NS, aes(x=Age, y=Alcohol)) + geom_point()

  1. Which variables listed in the dataset are significantly related to plasma beta carotene?
numerical_vars <- c("Age", "Quetelet", "Calories", "Fat", "Fiber", "Alcohol", "Cholesterol")
for(var in numerical_vars) {
  cat("Testing", var, ":\n")
  cor_test <- cor.test(NS[[var]], NS$BetaPlasma, method = "pearson")
  cat("Correlation Coefficient:", cor_test$estimate, "\n P-value:", cor_test$p.value, "\n\n")
}
## Testing Age :
## Correlation Coefficient: 0.1011276 
##  P-value: 0.07308498 
## 
## Testing Quetelet :
## Correlation Coefficient: -0.2293877 
##  P-value: 3.957005e-05 
## 
## Testing Calories :
## Correlation Coefficient: -0.02220696 
##  P-value: 0.6946032 
## 
## Testing Fat :
## Correlation Coefficient: -0.09164659 
##  P-value: 0.1044788 
## 
## Testing Fiber :
## Correlation Coefficient: 0.2359536 
##  P-value: 2.326513e-05 
## 
## Testing Alcohol :
## Correlation Coefficient: -0.02221084 
##  P-value: 0.6945525 
## 
## Testing Cholesterol :
## Correlation Coefficient: -0.130305 
##  P-value: 0.02070275
categorical_vars <- c("Smoke", "Sex", "VitaminUse")
for(var in categorical_vars){
  cat("Testing", var, ":\n")
  if(length(unique(NS[[var]])) == 2) {
    t_test_result <- t.test(BetaPlasma ~ get(var), data = NS)
    cat("P-value for T-test:", t_test_result$p.value, "\n\n")
  } else{
    anova_result <- aov(BetaPlasma ~ get(var), data = NS)
summary_anova <- summary(anova_result)
cat("P-value for ANOVA:", summary_anova[[1]][["Pr(>F)"]][1], "\n\n")
  }
}
## Testing Smoke :
## P-value for T-test: 5.174922e-06 
## 
## Testing Sex :
## P-value for T-test: 0.04747518 
## 
## Testing VitaminUse :
## P-value for ANOVA: 6.376754e-05
  1. Is there sufficient evidence that the proportions of smoking status (prior-smoke)- Never, former and current are equal?
NS$PriorSmoke <- as.factor(NS$PriorSmoke)
smoking_status_table <- table(NS$PriorSmoke)
chi_square_test <- chisq.test(smoking_status_table)
cat("P-value:", chi_square_test$p.value, "\n")
## P-value: 1.78434e-14
cat("There is sufficient statistical evidence to reject that the proportions of smoking status are equal. \n")
## There is sufficient statistical evidence to reject that the proportions of smoking status are equal.
  1. Consider the following variables: fat grams consumed in a day, cholesterol consumed in mg per day, and age in years. Which of the above factors significantly affects the variable calories consumed in a day at a 5% level of significance?
model<- lm(Calories ~ Fat + Cholesterol + Age, data = NS)
summary_model <- summary(model)
print(summary_model)
## 
## Call:
## lm(formula = Calories ~ Fat + Cholesterol + Age, data = NS)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -690.7 -176.6  -47.0  155.8 3315.6 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 512.9462    86.5106   5.929 8.09e-09 ***
## Fat          16.2645     0.7925  20.523  < 2e-16 ***
## Cholesterol   0.4208     0.2015   2.089   0.0375 *  
## Age          -1.4205     1.3040  -1.089   0.2769    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 331.9 on 311 degrees of freedom
## Multiple R-squared:  0.7643, Adjusted R-squared:  0.762 
## F-statistic: 336.1 on 3 and 311 DF,  p-value: < 2.2e-16
if(summary_model$coefficients["Fat", "Pr(>|t|)"]< 0.05){
  cat("Fat grams consumed daily significantly affect calories consumed.\n")
}
## Fat grams consumed daily significantly affect calories consumed.
if(summary_model$coefficients["Cholesterol", "Pr(>|t|)"]< 0.05){
  cat("Mg of cholesterol consumed daily significantly affects calories consumed.\n")
}
## Mg of cholesterol consumed daily significantly affects calories consumed.
if(summary_model$coefficients["Age", "Pr(>|t|)"]< 0.05){
  cat("Age significantly affects calories consumed.\n")
}
cat("Age doesn't significantly affects calories consumed.\n")
## Age doesn't significantly affects calories consumed.
  1. Estimate the effect of smoking on cholesterol levels and provide a 95% confidence interval for it.
NS$Smoke <- as.factor(NS$Smoke)
model <- lm(Cholesterol ~ Smoke, data = NS)
summary(model)
## 
## Call:
## lm(formula = Cholesterol ~ Smoke, data = NS)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -200.01  -89.22  -35.51   66.18  662.99 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  237.707      7.983  29.777   <2e-16 ***
## SmokeYes      34.826     21.606   1.612    0.108    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 131.7 on 313 degrees of freedom
## Multiple R-squared:  0.008232,   Adjusted R-squared:  0.005064 
## F-statistic: 2.598 on 1 and 313 DF,  p-value: 0.108
ci <- confint(model, level = 0.95)
print(ci)
##                  2.5 %    97.5 %
## (Intercept) 221.999728 253.41351
## SmokeYes     -7.686032  77.33791
  1. Is the effect of smoking on cholesterol levels significant? Make your decision based on confidence interval and p-value.
NS$Smoke <- as.factor(NS$Smoke)
model <- lm(NS$Cholesterol ~ NS$Smoke)
summary_model <- summary(model)
print(summary_model)
## 
## Call:
## lm(formula = NS$Cholesterol ~ NS$Smoke)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -200.01  -89.22  -35.51   66.18  662.99 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  237.707      7.983  29.777   <2e-16 ***
## NS$SmokeYes   34.826     21.606   1.612    0.108    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 131.7 on 313 degrees of freedom
## Multiple R-squared:  0.008232,   Adjusted R-squared:  0.005064 
## F-statistic: 2.598 on 1 and 313 DF,  p-value: 0.108
ci <- confint(model, level = 0.95)
print(ci)
##                  2.5 %    97.5 %
## (Intercept) 221.999728 253.41351
## NS$SmokeYes  -7.686032  77.33791
coefficients_summary <- summary(model)$coefficients
smoking_effects <- coefficients_summary[coefficients_summary[, 4] < 0.05, ]
  1. Do the sample data provide evidence that vitaminuse affects retinol (micronutrient in the blood) levels? Justify your answer with a p-value.
NS$VitaminUse <- as.factor(NS$VitaminUse)
anova_result <- aov(RetinolPlasma ~ VitaminUse, data = NS)
summary_anova <- summary(anova_result)
print(summary_anova)
##              Df   Sum Sq Mean Sq F value Pr(>F)
## VitaminUse    2    21580   10790   0.246  0.782
## Residuals   312 13680538   43848
p_value <- summary(anova_result)[[1]]$`Pr(>F)`[1]
cat("P-value:", p_value, "\n")
## P-value: 0.7820154
cat("There is not statistically significant evidence to show that vitamin use affects retinol plasma levels.\n")
## There is not statistically significant evidence to show that vitamin use affects retinol plasma levels.
library(ggplot2)
ggplot(NS, aes(x = RetinolPlasma, fill = VitaminUse)) + geom_histogram(position = "dodge") + labs(title = "Histogram of Retinol Levels VS Vitamin Use", y = "Vitamin Use")
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

  1. Who (Man or Woman) is more likely to take a vitamin? Is there a significant association between vitamin use and the sex of the participants?
NS$VitaminUse <- as.factor(NS$VitaminUse)
NS$Sex <- as.factor(NS$Sex)
table_vitamin_sex <- table(NS$VitaminUse, NS$Sex)
chi_square_test <- chisq.test(table_vitamin_sex)
print(chi_square_test)
## 
##  Pearson's Chi-squared test
## 
## data:  table_vitamin_sex
## X-squared = 11.071, df = 2, p-value = 0.003944
cat("There is statistical significance between vitamin use and gender.\n")
## There is statistical significance between vitamin use and gender.
library(ggplot2)
ggplot(NS, aes(x = Sex, fill = VitaminUse)) + geom_bar(position = "fill") + scale_fill_manual(values = c("No" = "red")) + labs(title = "Vitamin Use VS Sex", x = "Sex", y = "Vitamin Use")