Problem 3 - Basic Statistics for Business and Economics with Student

Question

A sample of General Mills employees was studied to determine their degree of satisfaction with their present life. A special index, called the index of satisfaction, was used to measure satisfaction. Six factors were studied, namely, age at the time of first marriage \(\left(X_{1}\right)\) annual income \(\left(X_{2}\right)\) number of children living \(\left(X_{3}\right)\) value of all assets \(\left(X_{4}\right)\) status of health in the form of an index \(\left(X_{5}\right)\) and the average number of social activities per week-such as bowling and dancing \(\left(X_{6}\right)\) Suppose the multiple regression equation is: $$ \hat{Y}=16.24+0.017 X_{1}+0.0028 X_{2}+42 X_{3}+0.0012 X_{4}+0.19 X_{5}+26.8 X_{\theta} $$ a. What is the estimated index of satisfaction for a person who first married at \(18,\) has an annual income of \(\$ 26,500,\) has three children living, has assets of \(\$ 156,000,\) has an index of health status of \(141,\) and has 2.5 social activities a week on the average? b. Which would add more to satisfaction, an additional income of \(\$ 10,000\) a year or two more social activities a week?

Step-by-Step Solution

Answer

a. 498.736. b. Two more social activities add more to satisfaction.

Step 1: Identify Variables

Identify the values of all the variables for the given person: - Age at first marriage, \( X_1 = 18 \). - Annual income, \( X_2 = 26500 \). - Number of children, \( X_3 = 3 \). - Value of assets, \( X_4 = 156000 \). - Health status index, \( X_5 = 141 \). - Average number of social activities, \( X_6 = 2.5 \).

Step 2: Substitute Values into Equation

Substitute the identified values into the regression equation: \[ \hat{Y} = 16.24 + 0.017(18) + 0.0028(26500) + 42(3) + 0.0012(156000) + 0.19(141) + 26.8(2.5) \]

Step 3: Calculate Each Term

Calculate each term of the equation using the substituted values: - \(0.017 \times 18 = 0.306\) - \(0.0028 \times 26500 = 74.2\) - \(42 \times 3 = 126\) - \(0.0012 \times 156000 = 187.2\) - \(0.19 \times 141 = 26.79\) - \(26.8 \times 2.5 = 67\)

Step 4: Add up all Terms

Add up all the calculated terms to find the estimated index of satisfaction: \[ \hat{Y} = 16.24 + 0.306 + 74.2 + 126 + 187.2 + 26.79 + 67 = 498.736 \]

Step 5: Determine Impact of Additional Income

Calculate the increase in the estimated index of satisfaction due to \$10,000 additional income: \(0.0028 \times 10000 = 28\)

Step 6: Determine Impact of Additional Social Activities

Calculate the increase in the estimated index of satisfaction due to two more social activities: \(26.8 \times 2 = 53.6\)

Step 7: Compare Impacts

Compare the increase in satisfaction from additional income (28) to that from additional social activities (53.6). The additional social activities contribute more to the index of satisfaction.

Key Concepts

Satisfaction Index

Understanding the concept of a satisfaction index is crucial, especially when measuring subjective feelings like happiness or contentment. The satisfaction index aims to quantify levels of satisfaction or well-being by considering various factors.
Imagine you want to understand how happy a person is. It's not just a single element that determines this happiness. Several aspects of life come together, such as personal relationships, income, health, and social activities.
In the exercise, the satisfaction index is created from specific factors: age at marriage, income, number of children, total assets, health status, and social activities. Each factor contributes a distinct weight to the satisfaction score.

• Age at first marriage might reflect early life satisfaction or maturity.
• Income often correlates with financial security, providing comfort and opportunities.
• The number of children can affect one's sense of fulfillment or stress.
• Value of assets can indicate wealth and future security.
• Health status directly impacts daily life quality.
• Social activities represent engagement and community interaction.

By combining these insights, we can estimate a person's satisfaction numerically, as shown in the original solution. The calculated value reflects the interaction and combined influence of all these factors, offering a comprehensive satisfaction snapshot.

Educational Statistics

Educational statistics play an essential role in making informed decisions and predictions based on data trends and patterns. It involves analyzing data in a way that helps understand different phenomena, such as the satisfaction index in our exercise.
In the context of the provided solution, educational statistics help in understanding the distribution and impact of various life factors on overall satisfaction. These statistics allow researchers to:

• Identify trends among significant population groups, like employees.
• Analyze relationships between different variables which affect satisfaction.
• Create models that predict future scenarios based on historical data.

For example, statistical analysis helps to determine whether income or social activities have a stronger correlation with satisfaction. Educational statistics are powerful as they strip down subjective biases and reliance on assumptions, focusing purely on what the data reflects.
Moreover, learning to work with these tools adds critical thinking and analytical skills, invaluable in various fields like sociology, psychology, business, and of course, education itself.

Predictive Modeling

Predictive modeling is a method used to forecast outcomes by learning patterns from existing data. It's like piecing together a puzzle that can predict future events based on past data. In this exercise, the model used is a multiple regression analysis.
Multiple regression helps us understand the relationship between a single outcome and multiple predictors. Here, it predicts satisfaction index based on multiple life aspects.
This model estimates how each factor separately influences satisfaction while considering the impact of other factors. For instance, how much more satisfied would someone be if their income increased or their number of social activities went up?

• Predictive modeling offers insights about which variables hold significant impacts.
• It allows for comparison of different variables' effects, such as income vs. social activities.
• The outcome (satisfaction index) can help in planning future steps to enhance quality of life.
• Organizations can use this model to understand their workforce better, leading to improved policies or management practices.

In practical terms, predictive modeling involves using past data trends to form a picture of possible future outcomes, thus helping individuals and organizations make better-informed decisions.