## Question 1

**Problem Description
**

**Example A: **Ice Cream and Cognitive Development

In this STATA assignment, we investigate the relationship between ice cream consumption and cognitive development. We consider the hypothesis that giving kids dessert leads to increased happiness and, in turn, better cognitive development.

## Solution

- A positive correlation was found between ice cream consumption per capita (Y) and mean scores on a standardized reading test (X), suggesting that cognitive development improves as ice cream consumption increases.
- While there is a positive correlation between X and Y, it's crucial to note that correlation does not imply causation. Multiple factors, including happiness, well-being, and physical health, must be considered to establish a causal relationship between ice cream consumption and cognitive development.
- The hypothesis was partially correct, but the involvement of mediator variables, such as happiness and well-being, is essential to fully understand the relationship.

**Problem Description
**

**Example B: **Homicide Rates and Marriage

For the second example, we explore the relationship between homicide rates and the share of married women in a developing country. The hypothesis is that homicides lead to a decrease in the number of available men, resulting in fewer married women.

## Solution

- There was a negative relationship between homicide rates and the share of married women, suggesting that as homicide rates decrease, the share of married women increases, and vice versa.
- The hypothesis is questionable because it doesn't consider other mediator variables, such as the share of men, causes of homicides, employment, and economic status.
- There was no reverse causality among the variables, indicating that the hypothesis is valid, but additional mediator variables should be included.

## Question 2: Social Media and Academic Outcomes

**Problem Description**

This question addresses the impact of social media on academic outcomes. The data includes high school students' SAT scores and the time they spent on social media in the week before the exam.

## Solution

- A negative correlation was observed between time spent on social media (X) and SAT scores (Y), implying that more time on social media results in less time spent studying and poorer academic outcomes.
- Policies encouraging reduced social media use to improve academic outcomes are recommended. Educational seminars can also be organized to raise awareness about the effects of social media on studying.
- Peer influence, if used positively for academic discussions, can enhance academic performance.
- Gender can be considered as a mediator variable to examine the impact of social media on academic performance.
- The error term in the model (ui) accounts for unobserved variables that can affect outcomes.
- Potential variables in ui include peer influence, gender, and family problems.
- β ̂_0 suggests that a student with no social media experience the week before the SAT exam in 2015 has an estimated SAT score of 1780.
- If a student spent 3.5 hours on social media, their predicted SAT score would be 1297.
- The estimated β ̂_0 may not be reliable, as it exceeds the SAT score range (400-1600).
- Assumptions include a linear relationship between SAT scores and time on social media, normality of SAT scores, and time spent on social media.

## Question 3: Regression Analysis

**Problem Description**

This question involves a regression analysis with X and Y variables. The dataset is presented with various statistics.

## Solution

- The mean of X1 is 9, and the mean of Y1 is 7.5. The slope coefficient is 0.5.
- The ordinary least squares estimate of the intercept is 3.
- The least square estimate indicates that for every unit increase in X1, Y1 increases by 0.5, with an expected value of 3.0 when X1 is zero.
- Stata output is provided, showing model statistics.
- A mean and standard deviation table is presented for X1, Y1, X2, Y2, X3, Y3, X4, and Y4.
- Slopes and coefficients for the four pairs of X and Y variables are listed.

## Question 4: Firearms and Violence

**Problem Description**

This question explores the relationship between firearm possession and violence. Two regression models are considered.

## Solution

- The slope of the regression is expected to be positive, indicating that firearm possession aggravates anger and leads to violence.
- The slope of the regression is expected to be negative, as mutual firearm possession reduces the likelihood of violence.
- The ordinary least squares estimate for one regression model is H = 21.44 + 0.02G.
- An alternative least squares estimate for the other model is G = -34.96 + 3.72H.
- The first regression model demonstrates internal validity.
- The model also exhibits external validity and can be applied to the broader population.
- Additional information required includes crime history, drug use, and gang membership for firearm owners.
- A correlation analysis can be conducted to determine the direction of causality between the variables.