In this comprehensive statistical analysis using R, we delve into various domains to answer pertinent questions. Explore the effects of caffeine on respiratory exchange ratios (RER) during exercise and discover if insurance costs are predictable based on age, BMI, and the number of children. Investigate the performance of a new advertising campaign compared to the old one and understand how argument strengths influence perceptions of trust and connectedness on social media profiles. These analyses provide valuable insights for making informed decisions in healthcare, marketing, and social science, demonstrating the power of statistical tools in extracting meaningful information.

## Problem Description

This statistical analysis assignment addresses key questions through statistical analysis. Can caffeine significantly impact respiratory exchange ratios (RER) during exercise? Are insurance costs predictable based on factors like age, BMI, and the number of children? Does a new advertising campaign outperform the old one? Do argument strengths influence trust and connectedness perceptions on social media profiles? In this assignment, we'll be working with several datasets to perform statistical analyses. We'll answer various research questions using R, such as:

**Insurance Costs:**Can we predict insurance costs based on factors like age, BMI, and the number of children? Are there correlations between these variables?**Caffeine Impacts on Respiratory Exchange Ratio (RER):**Does caffeine consumption affect RER during exercise? We'll compare RER values between individuals who took caffeine and those who received a placebo.**Impact of Advertising:**Does a new advertising campaign perform significantly better than the old one? We'll compare the ratings of the new and old ads.**Perceptions of Social Media Profiles:**How do different argument strengths influence perceptions of trust, connectedness, and knowledge of a social media profile owner?

Now, let's present each analysis with proper formatting and explanations:

**The effects of caffeine on respiration (RER)**

**Caffeine Impacts on Respiratory Exchange Ratio (RER)**

A study aimed to investigate the effect of caffeine on muscle metabolism during arm exercise. Half of the participants received caffeine, while the other half received a placebo. The respiratory exchange ratio (RER) was measured during exercise. RER indicates whether energy is obtained from carbohydrates or fats.

**R - code**

```
summary(RespiratoryExchangeSample)
# Perform a t-test to compare RER between the placebo and caffeine groups
t.test(Placebo, Caffeine)
```

The mean RER for the placebo group was 90.11, while the caffeine group had a mean RER of 110.85. The t-test results showed that there is a significant difference between the means, suggesting that caffeine impacts RER during exercise.

**Insurance Costs**

**Insurance Costs and Correlations**

We want to predict insurance costs based on age, BMI, and the number of children. Additionally, we aim to explore correlations between these variables.

**R- code**

```
Insurance2 <- data.frame(age, bmi, children, charges)
Corr_matrix <- cor(Insurance2)
library(corrplot)
corrplot(Corr_matrix, type = "upper", order = "hclust", col = brewer.pal(n = 8, name = "RdYlBu"))
# Perform a regression analysis to predict insurance costs
model1 <- lm(charges ~ age + bmi + children, data = Insurance2)
summary(model1)
```

The correlation analysis indicates weak positive correlations among the variables. The regression analysis shows that all the variables (age, BMI, and children) are statistically significant, with age having the largest impact on insurance costs.

**Incorporating Gender and Smoking Status**

We further enhance the model by incorporating gender and smoking status (smoker2).

```
Insurance <- mutate(Insurance, gender = ifelse(sex == "female", 1, 0))
Insurance <- mutate(Insurance, smoker2 = ifelse(smoker == "yes", 1, 0))
Insurance2 <- data.frame(age, bmi, children, charges, gender, smoker2)
# Perform regression analysis with gender and smoking status
model2 <- lm(charges ~ age + bmi + children + gender + smoker2, data = Insurance2)
summary(model2)
```

The extended model includes gender and smoking status, with the results indicating that gender does not significantly impact costs, while smoking status does.

**Impact of Advertising**

**Impact of New Advertising Campaign**

To assess the impact of a new advertising campaign compared to the old one, we analyze the ratings given by two groups.

**R- code**

```
summary(Advertising)
# Perform a t-test to compare ratings between the two advertising groups
t.test(Rating ~ Group)
```

The analysis suggests that the mean rating for the new advertising campaign (52.34) is not significantly different from the old campaign (49.81), implying that the new campaign may not perform significantly better.

**ANOVA Analysis**

**Perceptions of Social Media Profiles**

We examine how different argument strengths influence perceptions of trust, connectedness, and knowledge of social media profile owners using ANOVA.

**R- code**

```
a <- aov(Trust ~ Argument, data = Perceptions)
b <- aov(Connectedness ~ Argument, data = Perceptions)
c <- aov(Knowledge ~ Argument, data = Perceptions)
summary(a); summary(b); summary(c)
# Perform post-hoc Tukey tests to identify differences
TukeyHSD(a)
TukeyHSD(b)
```

The ANOVA analysis shows that trust and connectedness are significantly different across argument strengths, while knowledge is not. Post-hoc tests reveal that the mean trust and connectedness for weak arguments differ from average and strong arguments.

These statistical analyses provide valuable insights into various research questions and help make informed decisions in different domains, from healthcare to marketing and social science.

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