Decide Whether You Would Expect Relationships To Be

Decide Whether You Would Expect Relationships Be

Analyze the expected relationships between various pairs of dependent and independent variables, providing explanations for whether these relationships are positive, negative, or ambiguous.

a. The amount of hair on the head of a male professor and the age of that professor.

b. Aggregate bet investment in China in a given year and the GDP in that year.

c. The growth rate of GDP in a year and the average hair length in that year.

d. The quantity of canned tuna demanded and the price of a can of tuna.

Paper For Above instruction

The relationships between variables can often provide insights into underlying causal mechanisms and economic behaviors. Analyzing each pair of variables involves considering potential correlations, causal directions, and external factors that might influence these relationships. The following discussions evaluate the expected nature of these relationships, whether positive, negative, or ambiguous, based on economic theory, biological considerations, and empirical evidence.

a. The amount of hair on the head of a male professor and the age of that professor

The relationship between age and hair amount in males is well-documented in biological and dermatological studies. As men age, they often experience hair thinning and balding due to hormonal changes and genetic factors. Typically, younger males tend to have fuller hair, while older males are more likely to have less hair or bald patches. Therefore, one would expect a negative relationship between age and hair volume — as age increases, hair volume decreases. This relationship is generally consistent and predictable, although individual variation exists due to genetics or health issues. Consequently, this relationship is expected to be negative.

b. Aggregate bet investment in China in a given year and the GDP in that year

Bet investments, especially in sports betting or casino gambling, are often sensitive to economic conditions. During times of economic expansion, disposable income tends to increase, potentially leading to higher gambling expenditures, which can serve as a form of entertainment or risk-taking behavior driven by increased wealth. Conversely, during economic downturns, consumer spending on discretionary activities like betting may decline. Empirical research suggests a positive correlation between economic growth (reflected in GDP) and gambling expenditure, as higher income levels enable more discretionary spending. Thus, a positive relationship between aggregate bet investment and GDP is expected, with higher GDP associated with increased betting activity.

c. The growth rate of GDP in a year and the average hair length in that year

This relationship is more ambiguous. The economic growth rate reflects macroeconomic performance, while average hair length is a biological trait influenced primarily by genetics, health, and cultural factors, not economic conditions. Although wealth and economic prosperity can influence lifestyle choices and grooming habits, the direct impact on hair length is minimal or indirect at best. For example, wealthier societies might have better access to hair care products or grooming services that influence hair length or style, but this effect is not guaranteed nor necessarily systematic. Therefore, the relationship is ambiguous and unlikely to be strongly correlated—making it uncertain whether economic growth influences hair length significantly.

d. The quantity of canned tuna demanded and the price of a can of tuna

The law of demand in economics states that, ceteris paribus, as the price of a good increases, the quantity demanded decreases — indicating a negative relationship. Consumers typically buy less tuna when prices rise due to substitution effects and income effects, and more when prices fall. Therefore, we expect a negative relationship between the price of tuna and its quantity demanded. This relationship is well-established and consistent across markets for most goods, including canned tuna.

Analysis of a height/weight regression example

The height versus weight example illustrates regression analysis's practical application. Typically, when fitting a regression line, some data points deviate significantly from the predicted values. The three most distant points—outliers—represent individuals whose observed weights differ markedly from their predicted weights based on height. Including these outliers can distort the regression estimates, bias the coefficients, and reduce model accuracy. Dropping such outliers often improves the model's fit by reducing heterogeneity and emphasizing the core relationship. However, removing data points must be justified carefully to avoid biasing results or losing important variability.

The regression model predicts the same weight for all males of the same height because it estimates an average relationship across the sample population. This is a feature of the linear regression model—predictions are mean responses conditioned on specific independent variable values. Since individual differences—such as muscle mass, age, health, or genetics—are not captured explicitly, the model cannot account for variability within each height group, leading to identical predicted weights for individuals with the same height.

Relationship between house size and price

The given regression equation is: SIZEi = -290 + 3.62 PRICEi, where SIZEi indicates house size in square feet and PRICEi indicates house price in thousands of dollars. The coefficient 3.62 on PRICEi suggests that for each additional thousand dollars in house price, the house size increases by approximately 3.62 square feet. The intercept -290 indicates the predicted house size when the price is zero, which is not meaningful in practice but necessary mathematically.

When evaluating the model with an R² of 0.4, it indicates that 40% of the variation in house size can be explained by the house price, with 60% due to other factors or random variation. While not a perfect fit, this level of explanatory power suggests moderate predictive ability. It might warrant incorporating additional variables, such as location, number of bedrooms, or lot size, to improve accuracy.

If house prices were measured in dollars instead of thousands of dollars, the coefficient would be scaled accordingly. Specifically, because the model estimates the relationship per thousand dollars, changing to dollars would increase the coefficient by a factor of 1,000. In other words, the new coefficient would be 3.62 × 1,000 = 3,620, meaning that for each additional dollar in house price, the predicted size increases by about 3.62 square feet. The intercept would also be adjusted appropriately, reflecting the scale change.

Estimating the relationship between agricultural employment percentage and per capita income

Given data on the percentage of the labor force employed in agriculture (AGR) and per capita income (PCGDP) for ten developed countries, the goal is to estimate the regression coefficients (β0 and β1). Using the formulas for simple linear regression, coefficients are calculated by:

• β1 = Cov(AGR, PCGDP) / Var(AGR)
• β0 = Mean(PCGDP) - β1 × Mean(AGR)

Calculating these requires the mean, variance, and covariance of the variables. Supposing we have the data for each country, the calculations involve first computing the means of AGR and PCGDP, then the variance of AGR and the covariance between AGR and PCGDP. For example, if the mean AGR is 30% and the mean PCGDP is $20,000, with calculated covariance and variance from the data, the regression slope β1 indicates how much per capita income changes with a percentage point change in agricultural employment. Typically, in developed countries, we expect an inverse relationship: higher agricultural employment tends to associate with lower per capita income, implying a negative slope.

The R² value measures the proportion of variability in per capita income explained by agricultural employment. Adjusted R² accounts for sample size and the number of predictors, providing a more accurate estimate of the model's explanatory power, especially with small samples like ten countries.

Plotting these data points on a graph with AGR on the x-axis and PCGDP on the y-axis, along with the regression line, helps visualize the fit. A downward-sloping line would support the hypothesis that increased agricultural employment correlates with lower income levels in developed countries.

Conclusion

These analyses collectively demonstrate the importance of understanding variable relationships, model interpretation, and statistical estimation in economics. Properly identifying the direction and nature of relationships enables better policymaking, forecasting, and understanding of market behaviors. Regression analysis, as exemplified in the height/weight and house price models, provides valuable tools for quantifying these relationships and assessing model adequacy through measures like R². Accurate interpretation of coefficients and model fit is essential for drawing meaningful economic insights.

References

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