Mastering Statistics: A Comprehensive Assignment on Least Square Estimates and Statistical Analysis

Problem Description:

This assignment delves into statistical analysis, particularly focusing on the application of least square estimates and statistical methods. It comprises a series of questions that challenge your ability to calculate predictions, assess the significance of coefficients, and make informed decisions in the world of data and statistics. From hypothesis testing to real-world economic implications, these questions cover a broad spectrum of statistical concepts to sharpen your analytical skills.

Question 1: Best Linear Unbiased Prediction

In this problem, we encounter a model y = β1 + β2 x + ε, estimated through the least squares method, with specific results at hand. Our objective is to calculate the best linear unbiased prediction of y when x = 5.

Options:

1. 105.5.
2. 21.1.
3. 71.1.
4. 26.1.

Question 2: Computing Ordinary R2 Statistic

This question revolves around a multiple regression model with provided ANOVA table data. The task is to compute the ordinary R2 statistic.

Suppose a multiple regression model

y = β1 + β2 x2 + ... + βK xK + ε

ANOVA | Degrees of freedom | Sum of squares | Mean squares

Explained | 6 | 22 | 3.667

Residual | 55 | 12 | 0.218

Total | 61 | 34 | 0.557

Use the information in the table to compute the ordinary R2 statistic.

1. R2 = 0.647 .
2. R2 = 0.560 .
3. R2 = 0.220 .
4. R2 = 0.609 .

Question 3: T-stat and Null Hypothesis

A regression has been conducted with 5 variables on Y, based on 24 observations, and the t-statistic yields 2.59. We must decide what to conclude about the null hypothesis at a 95% confidence level.

Options:

1. Fail to reject the null hypothesis since t is less than the critical value.
2. Fail to reject the null hypothesis since t is greater than the critical value.
3. Reject the null hypothesis in favor of the alternative since t is greater than the critical value.
4. Reject the null hypothesis in favor of the alternative since t is less than the critical value.

Question 4: Computing F-statistic

Given another multiple regression model with ANOVA table data, we are tasked with calculating the F-statistic for testing the null hypothesis that all the slope coefficients are zero.

Suppose a multiple regression model

y = β1 + β2 x2 + ... + βK xK + ε

ANOVA | Degrees of freedom | Sum of squares | Mean squares

Explained | 2 | 178 | 89.00

Residual | 87 | 135 | 1.552

Total | 89 | 313 | 3.517

Compute the F-statistic for testing the null hypothesis. That is, that 0 = β2 = ... = βK.

1. F = 57.356 .
2. F = 89.00 .
3. F = 3.52 .
4. F = 1.55 .

Question 5: Understanding Coefficient Effects

You conducted a regression with GDP on Federal Funds Rate and Rainfall, resulting in specific coefficients. Your mission is to discern the impact of a one-inch decrease in rainfall.

Options:

1. GDP would fall by 5m USD (5,000,000) holding the federal funds rate constant.
2. GDP would decrease by 23m USD (23,000,000).
3. GDP would decrease by 23m USD (23,000,000) holding all else constant.
4. GDP would fall by 5m USD (5,000,000).

Question 6: Sales and Advertising Expenditures

This comprehensive question pertains to the relationship between sales and advertising expenditures, encompassing calculations for degrees of freedom, hypothesis testing, predictions, and evaluating the economic significance of advertising on sales.

Assume the error term in the true population regression function is normally-distributed, but note the very small sample size.

Sales = 550.0 (6.5) + 11.0 Advertising expenditures (4.0)

1. For purposes of computing confidence intervals or tests, what are the degrees of freedom for this problem? (Give a number.) n-2 = 11
2. Given the test statistic, the critical value of the t-stat for a 95% confidence level or the p-value, and your conclusion (whether the null hypothesis can be rejected). T= b/SE = 6.5/11 =0.59 and since the slope is not equal to zero, there is a significant relationship between the predictor and response variable
3. Suppose advertising expenditures were zero. Give a prediction for sales. The sales would be 3575 million dollars
4. Suppose advertising expenditures were increased by 5 million dollars. Would sales likely increase or decrease? Give a prediction for the change in sales. If expenditures were increased, sales would increase to a tune of 3630 million dollars
5. In the context of the question, is advertising economically significant?

Yes advertising is economically significant because it brings an increase to the sales

Question 7: Asynchronous and Synchronous Components

This question seeks your perspective on what you appreciate about both the asynchronous and synchronous components of the course.

What is one thing you like about the asynchronous component and synchronous component of the course (one for each)?

The asynchronous component I like about the course is that we can view instructional materials each week anytime a person wishes to. The synchronous component I like about the course is that someone can log in and participate in class activities anytime a person wishes to.

Question 8: Suggestions for Improvement

Here, you are invited to share your thoughts on what changes you would introduce to enhance both the asynchronous and synchronous components of the course and improve the overall learning experience.

What is one thing you would change about the asynchronous component and synchronous component of the course (one for each)?

One thing I would change about the asynchronous component of the course is the availing all the course materials at once from the beginning of the semester. On the synchronous component of the course, I would make the login activity accessible from every part of the world.