91Ó°ÊÓ

Researchers developed a model to explain the age gap between husbands and wives at first marriage. The model is below: $$ \hat{y}=0.0321 x_{1}+0.9848 x_{2}+0.5391 x_{3}-0.000145 x_{4}^{2}+3.8483 $$ where y: Age gap at first marriage (male - female) \(x_{1}:\) Percent of children aged 10 to 14 involved in child labor \(x_{2}:\) Indicator variable where 1 is an African country, 0 otherwise \(x_{3}:\) Percent of the population that is Muslim \(x_{4}:\) Percent of the population that is literate Source: Xu Zhang and Solomon W. Polachek, State University of New York at Binghamton "The Husband- Wife Age Gap at First Marriage: A Cross-Country Analysis" (a) Use the model to predict the age gap at first marriage for an African country where the percent of children aged 10 to 14 who are involved in child labor is \(12,\) the percent of the population that is Muslim is \(30,\) and the percent of the population that is literate is \(75 .\) (b) What would be the mean difference in age gap between an African country and a non-African country? (c) Interpret the coefficient of "percent of children aged 10 to 14 involved in child labor." (d) The coefficient of determination for this model is 0.593 . Interpret this result. (e) The \(P\) -value for the test \(H_{1}: \beta_{1} \neq 0\) versus \(H_{1}: \beta_{1} \neq 0\) is \(0.008 .\) What would you conclude about this test?

Step by step solution

01

Title - Predict Age Gap (a)

Substitute the given values into the model. For an African country where \(x_{1}=12\), \(x_{2}=1\), \(x_{3}=30\), \(x_{4}=75\), calculate \(\hat{y}=0.0321 \times 12+0.9848 \times 1+0.5391 \times 30-0.000145 \times 75^{2}+3.8483\).

02

Title - Calculate Age Gap

Evaluate each term: 1. \(0.0321 \times 12 = 0.3852\), 2. \(0.9848 \times 1 = 0.9848\), 3. \(0.5391 \times 30 = 16.173\), 4. \(-0.000145 \times 75^{2} = -0.814375\). Add these values to the constant term 3.8483 to get the predicted age gap.

03

Title - Total Prediction

Sum the results from Step 2: \(0.3852 + 0.9848 + 16.173 - 0.814375 + 3.8483\). The predicted age gap is approximately \(20.577\).

04

Title - Mean Difference in Age Gap (b)

The mean difference in age gap between an African country and a non-African country is determined by the coefficient \(x_{2}\), which is 0.9848. Since \(x_{2}\) is 1 for African countries and 0 for non-African countries, the difference in age gap is \(0.9848 - 0 = 0.9848\) years.

05

Title - Interpretation of Coefficient (c)

The coefficient of 'percent of children aged 10 to 14 involved in child labor' is 0.0321. This means that for every 1% increase in child labor involvement, the age gap at first marriage increases by approximately 0.0321 years, assuming all other variables are held constant.

06

Title - Coefficient of Determination (d)

The coefficient of determination, \(R^2\), is 0.593. This means that approximately 59.3% of the variation in the age gap at first marriage can be explained by the model using the given variables. The remaining 40.7% is explained by other factors not included in the model.

07

Title - P-value Interpretation (e)

The P-value for the test \(H_{1}: \beta_{1} eq 0\) is 0.008. Since this value is less than the typical significance level of 0.05, we reject the null hypothesis and conclude that the coefficient for percent of children aged 10 to 14 involved in child labor is statistically significant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Regression Analysis

Regression analysis is a powerful statistical method used to understand and quantify relationships between variables. In this exercise, the researchers developed a regression model to predict the age gap at first marriage using various factors:

1. Percent of children aged 10 to 14 involved in child labor
2. Whether the country is African
3. Percent of the population that is Muslim
4. Percent of the population that is literate

The model takes the form of a linear equation, where coefficients are assigned to each variable, reflecting their impact on the age gap. By plugging in specific values for these variables, we can predict the age gap between husbands and wives in different contexts.

Age Gap Prediction

In this exercise, we use the given regression model to predict the age gap at first marriage. Given the values for an African country (12% child labor, 1 for African country, 30% Muslim population, 75% literate), we substitute these values into the model:

\beginenvangels: \(hat{y} = 0.0321 \times 12+0.9848 \times 1+0.5391 \times 30-0.000145 \times 75^{2}+3.8483\)This results in a predicted age gap of approximately 20.577 years. Each value is substituted and calculated step-by-step to arrive at this prediction.

Statistical Significance

Statistical significance helps us determine if the relationships in the model are meaningful. In this exercise, the P-value for the variable 'percent of children aged 10 to 14 involved in child labor' is 0.008. Since this P-value is less than the conventional threshold of 0.05, we can reject the null hypothesis that this variable has no effect. This indicates that the coefficient is statistically significant, meaning child labor involvement has a meaningful impact on the age gap at first marriage.

Coefficients Interpretation

Understanding coefficients is crucial in regression analysis. Each coefficient represents the expected change in the dependent variable (age gap) for a one-unit change in the predictor, holding other variables constant.

For example, the coefficient for 'percent of children aged 10 to 14 involved in child labor' is 0.0321. This tells us that for every 1% increase in child labor, the age gap increases by 0.0321 years, assuming other variables remain unchanged.

Coefficient of Determination

The coefficient of determination, denoted as \(R^2\), measures how well the regression model explains the variation in the dependent variable. An \(R^2\) value of 0.593 means that 59.3% of the variation in the age gap at first marriage can be explained by the variables in the model. The rest, 40.7%, is attributed to factors not included in the model. High \(R^2\) values indicate a good fit, but even lower values can be meaningful depending on the context and complexity of the subject matter.