Part A: Exploratory Data Analysis
In this section, we will conduct exploratory data analysis on a dataset containing information about individuals’ happiness levels. The dataset includes variables such as age, income, marital status, and happiness score.
First, we will load the dataset into R and examine its structure, dimensions, and variable types. We will use the `read.csv()` function to read the data from a CSV file into a data frame. Then, we will use the `str()` function to view the structure of the data frame, the `dim()` function to check the dimensions of the data frame, and the `class()` function to determine the variable types.
Next, we will calculate summary statistics for the continuous variables in the dataset, such as age and income. We will use the `summary()` function to obtain measures such as the mean, median, and standard deviation of these variables. Additionally, we will create histograms to visualize the distribution of these variables using the `hist()` function.
Afterwards, we will explore the relationships between the variables in the dataset by creating scatterplots and calculating correlation coefficients. We will use the `plot()` function to create scatterplots and the `cor()` function to calculate correlation coefficients. By examining the scatterplots and correlation coefficients, we can determine if there are any significant relationships between variables.
Finally, we will perform a simple linear regression analysis to predict the happiness score based on a selected independent variable. We will use the `lm()` function to fit a linear regression model and the `summary()` function to obtain the regression coefficients, standard errors, and R-squared value. This analysis will help us understand the relationship between the independent variable and the happiness score.
Part B: Inferential Statistics
In this section, we will perform inferential statistical analysis on the dataset to draw conclusions about the population based on the sample data. We will focus on hypothesis testing and confidence intervals.
First, we will formulate null and alternative hypotheses to test a specific research question. For example, we might be interested in testing whether there is a significant difference in happiness levels between married and unmarried individuals. We will use the appropriate statistical test, such as a t-test or chi-squared test, to test the null hypothesis.
Next, we will calculate the test statistic and p-value based on the sample data. We will use functions such as `t.test()` or `chisq.test()` to perform the hypothesis test and obtain the test statistic and p-value.
To interpret the results, we will compare the p-value to the significance level (e.g., α = 0.05) and make a decision to either reject or fail to reject the null hypothesis. If the p-value is less than the significance level, we will conclude that there is sufficient evidence to support the alternative hypothesis.
Additionally, we will calculate confidence intervals for population parameters, such as the mean happiness score or the difference in mean scores between two groups. We will use functions like `t.test()` or `prop.test()` to calculate the confidence intervals.
By examining the confidence intervals, we can estimate the range within which the population parameter is likely to fall. If the confidence interval does not include a specified value, we can make conclusions about the population parameter based on the sample data.
Part C: Mixed ANOVA
In this section, we will conduct a mixed analysis of variance (ANOVA) to examine the effects of two independent variables, age group and marital status, on the dependent variable, happiness score. Mixed ANOVA can be used when we have both between-subjects and within-subjects factors.
First, we will define the factors and levels of our design. In this case, age group has three levels (e.g., young, middle-aged, old) and marital status has two levels (e.g., married, unmarried). The dependent variable, happiness score, will be measured at multiple time points.
Next, we will check the assumptions of the mixed ANOVA, such as normality, homogeneity of variance, and sphericity. We will use graphical methods like Q-Q plots and statistical tests like Levene’s test or Mauchly’s test to assess these assumptions.
Then, we will perform the mixed ANOVA using the appropriate statistical procedure, such as the `aov()` or `ezANOVA()` function. We will include the main effects of age group and marital status, as well as their interaction, in the analysis.
After conducting the mixed ANOVA, we will interpret the results by examining the main effects and interaction effects. We will focus on the significance levels (e.g., p < 0.05) and effect sizes (e.g., eta-squared) to determine the strength and direction of the effects. Finally, we will post-hoc analyses to further explore significant effects. We might use procedures like Tukey's HSD test or pairwise t-tests to compare means between specific groups and investigate any significant differences. In conclusion, this assignment provides an opportunity to apply various statistical techniques to explore and analyze a dataset. By conducting exploratory data analysis, inferential statistics, and mixed ANOVA, we can gain insights into the relationships between variables, conduct hypothesis tests, calculate confidence intervals, and examine the effects of independent variables on the dependent variable. These analyses can help inform decision-making and draw conclusions based on empirical evidence.