Question 1: Measures of Central Tendency and Dispersion for Select Demographic Variables in Lebanon

The following table shows the descriptive statistics for the monthly births, marriages and divorces in Beirut, Lebanon, over the period 2008 2010.

Table 1: Descriptive Statistics: Beirut births, marriages and divorces over the period 2008 2010

Measure Beirut Births Beirut Marriages Beirut Divorces

Mean 936.69 373.97 78.08

Median 821 353.5 77.5

Mode 759 306 84

Standard deviation 330.2615 76.0163 11.4776

Source: Central Administration of Statistics, Lebanon

From the preceding table, the mean monthly births in the Beirut region stood at 936 during the period 2008 2010, while marriages and divorces averaged 373 and 78 respectively, per month. Between 2008 and 2010, the median monthly births in Beirut were 821, while the median marriages and divorces per month was 353 and 77 respectively. In the Beirut region, most months registered 759, 306, and 84 births, marriages, and divorces respectively over the period 2008 -2010. The standard deviation of monthly births, marriages, and divorces in Beirut was 330.2615, 76.0163, and 11.4776 respectively.

Question 2: Correlation Coefficient and Coefficient of Determination

Table 2 below shows the coefficient of the correlation between the variables presented in table 1 above.

Table 2: Correlation matrix

Coefficient of correlation Total Beirut Births Beirut Marriages Beirut Divorces

Total Beirut Births 1 Beirut Marriages 0.3000 1 Beirut Divorces 0.0246 0.3815 1

The coefficient of the correlation between monthly Beirut births and the monthly Beirut marriages is 0.3, suggesting that these variables have a weak correlation. The monthly Beirut marriages also have a low correlation with the monthly Beirut divorces because the correlation coefficient is 0.3815. The weakest correlation is the one between the monthly Beirut births and the monthly Beirut divorces; the correlation coefficient of these two variables is 0.0246, which is the least coefficient in table 2 above. The table 3 below shows the coefficient of determination.

Table 3: Coefficient of Determination

Total Beirut Births Beirut Marriages Beirut Divorces

Total Beirut Births 1 Beirut Marriages 0.0900 1.0000 Beirut Divorces 0.0006 0.1455 1

The monthly Beirut births and the monthly Beirut marriages have a coefficient of determination of 0.09, which means these variables share 9% of their variation over the period 2008 2010. The monthly Beirut births and the monthly Beirut divorces have a coefficient of determination of 0.0006, suggesting that approximately 0.06% of the variation between these variables over the three-year period to 2010 is common. The monthly Beirut marriages and the monthly Beirut divorces have a coefficient of determination of 0.1455, which means that about 14.55% of the variation in these variables between 2008 and 2010 is common.

Table 3: Coefficient of Determination

Coefficient of determination Total Beirut Births Beirut Marriages Beirut Divorces

Total Beirut Births 1 Beirut Marriages 0.0900 1 Beirut Divorces 0.0006 0.1455 1

Question 3: Multiple Linear Regression Equation

The dependent variable in the regression equation is the total Beirut births, while the independent variables are the Beirut marriages, Beirut divorces, Beirut female deaths and Beirut male deaths. The following table summarizes the regression of the dependent variable on the four independent variables.

Table 4: Regression Summary

Regression Statistics

Multiple R 0.3669

R Square 0.1346

Adjusted R Square 0.0229

Standard Error 326.4501

Observations 36

The table shows that the multiple R is 0.3669, and this suggests that there is a low correlation between the predicted values and the actual values. The R Square of 0.1346 suggests that the model explains about 13.46% of the variation in the dependent variable. The Adjusted R Squared is 0.0229. The following table shows the results of the test of the significance of the regression model.

Table 5: Test of Model Significance

ANOVA df SS MS F Significance F

Regression 4 513884.0050 128471.0012 1.2055 0.3283

Residual 31 3303659.6339 106569.6656 Total 35 3817543.6389

The regression model was tested for significance using the F test of significance of a regression model; this test compares the mean regression sum of squares to the residual mean sum of squares in order to establish if the error in estimating the effect of a dependent variable exceeds the estimated effect. From table 5, the regression mean sum of squares is 128471, while the residual mean sum of squares is 106569; this results in an F ratio of 1.2055. After determining the F ratio, the next step entails finding out its probability of occurrence. The F test statistic, of which the F ratio is a variant, follows the F distribution, and it is from this distribution that one can get the probability of observing a test statistic of a particular value.

In testing the significance of the multiple regression model, the hypothesis was that, if the multiple regression model does not predict the dependent variable, there is at least a 5% (0.05) chance of observing a test statistic as large as 1.2055. At 4 and 31 degrees of freedom, the probability of observing a test statistic as large as 1.2055 is 0.3283, which is higher than the 0.05 probability that had been hypothesized. Therefore, there is a high probability that if the regression model does not significantly predict the dependent variable, one would observe a test statistic of 1.2055. Consequently, the relationship that the regression model describes is not significant. The following table shows the coefficient of the regression model.

Table 6: Regression Coefficients

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 216.9927 581.3789 0.3732 0.7115 -968.7373 1402.7227

Beirut marriages 1.4058 0.8112 1.7329 0.0930 -0.2487 3.0604

Beirut divorces -4.0070 5.2765 -0.7594 0.4534 -14.7686 6.7546

Beirut male deaths 0.8775 3.9249 0.2236 0.8246 -7.1273 8.8823

Beirut female deaths 3.0835 3.6405 0.8470 0.4035 -4.3414 10.5084

The predicted value of Beirut marriages for n+1 (where n is the most recent year) is 993 marriages. The t-test of significance of a regression coefficient was applied in testing the significance of the regression coefficients the test was performed at the 0.05 significance level. Looking at the p-values of all the coefficients of the regression model, it is apparent that none of them falls below 0.05, and this means they are not statistically significant.

Question 4: Linear and Quadratic Time Series Regressions for Beirut Marriages

a) Linear time series regression

The following table shows the summary of the linear time-series regression model for Beirut marriages over the period 2008 2010.

Table 7: Summary of Regression Model

Regression Statistics

Multiple R 0.0861

R Square 0.0074

Adjusted R Square -0.0218

Standard Error 76.8397

Observations 36

From the preceding table, the R Square is very low at 0.0074, it means the regression model explains only 0.74% of the variation in the dependent variable. The following table shows the results of the test of significance of the linear time-series regression model.

Table 8: Test of Significance: Linear Time-series Regression Model

ANOVA df SS MS F Significance F

Regression 1 1499.3519 1499.3519 0.2539 0.6176

Residual 34 200747.6203 5904.3418 Total 35 202246.9722

From the table, the residual mean squared errors for the linear time-series regression model is 5904.3418. Table 8 also shows that the linear time-series regression model is not significant because the F test statistic has a significance value of 0.6167, and this is way higher than the 0.05 level at which the regression model was tested for significance. Table 9 below shows the test of significance of the coefficient of the linear, time-series regression model.

Table 9: Test of Regression Coefficient for the Linear Time-Series Model

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 362.4794 26.1563 13.8582 0.0000 309.3233 415.6354

Month 0.6212 1.2328 0.5039 0.6176 -1.8841 3.1266

From the intercept and the coefficient of this model, the estimated Beirut marriages for n + 2 are 386.09. It is also notable that the intercept of the linear time-series regression model is significant (p-value falls below 0.05).

b) Quadratic time series regression

The following table summarizes the quadratic time-series regression model.

Table 10: Quadratic Time-series Regression Model

Regression Statistics

Multiple R 0.0917

R Square 0.0084

Adjusted R Square -0.0517

Standard Error 77.9560

Observations 36

Just as the linear time-series regression model, the quadratic time-series regression model explains very little of the variation in the dependent variable; the R Square is 0.0084, suggesting that the model explains about 0.84% of the variation in the dependent variable. The following table shows the residual mean squared errors.

Table 11: Sum of Squares and Model Significance

ANOVA df SS MS F Significance F

Regression 2 1701.2892 850.6446 0.1400 0.8699

Residual 33 200545.6830 6077.1419 Total 35 202246.9722

From table 11, the residual mean squared errors are 6077.1419. The table also shows that the test statistic F has a significance value of 0.8699, which means the quadratic time-series regression model does not significantly predict the dependent variable. Table 12 below shows the coefficients of the quadratic time-series regression model.

Table 12: Coefficients Quadratic Time-series Regression Model

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 356.7228 41.2484 8.6482 0.0000 272.8024 440.6433

Month 1.5302 5.1407 0.2977 0.7678 -8.9286 11.9889

Month Squared -0.0246 0.1348 -0.1823 0.8565 -0.2987 0.2496

From the coefficients of the regression model, it is estimated that Beirut marriages during time n+2 will be 413. Like in the linear time-series regression model, the intercept of the quadratic time-series model is significant because it has a p-value of 0. However, the other two coefficients are not statistically significant.

Question 5: Analysis of Regression Results

The multiple regression model has showed that marriages, divorces, female deaths and male deaths are not significant predictors of the births in the Beirut region of Lebanon; the test statistic F had a significance value exceeding 0.05 0.05 is the significance level at which the regression model was tested for significance. When regression models are not significant, it suggests there might not be a conceptual or theoretical link between the variables in the model (Brase and Brase, 2017; Ott and Longnecker, 2015). Thus, there is no conceptual link between the monthly births in Beirut and the marriages, divorces, female deaths and male deaths in Beirut. The time-series regression models have also shown that time is not a significant predictor of the monthly marriages in Beirut. Time-series models that do not control for extraneous variables are unlikely to explain the variation in the dependent variable (Jackson, 2015, Bryman, 2015). Therefore, the linear and quadratic time-series regression models are insignificant predictors of the monthly marriages in Beirut because they do not include extraneous variables that can account for the monthly variation in marriages in Beirut.

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References

Brase, C.H. and Brase, C.P. (2017). Understandable statistics: concepts and methods (11th edition). Boston, M.A: Cengage Learning.

Bryman, A. (2015). Social research methods. London: Oxford University Press.

Jackson, S. L. (2015). Research methods and statistics: A critical thinking approach. South Western: Cengage Learning.

Ott, R.L. and Longnecker, M. (2015)....

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