Regression Analysis and Statistical Inference: A Statistics Assignment

This assignment explores a multiple regression analysis, examining the relationship between the heights of sons, fathers, and mothers. The student begins by interpreting key statistical outputs, including the standard error of estimate and the coefficient of determination, to assess the model's accuracy and goodness of fit. The analysis continues with an interpretation of the adjusted coefficient of determination and a test of the overall utility of the model using an F-test. The student then interprets the coefficients of the regression model, explaining the impact of fathers' and mothers' heights on sons' heights. Finally, the assignment concludes by determining whether the data supports the inference of linear relationships between the heights of sons and fathers, and sons and mothers, providing a comprehensive understanding of the statistical analysis and its implications.

Running Head: STATISTICS 1
Topic: Statistics
By (Name of Student)
(Institutional Affiliation)

STATISTICS 2
1. To determine whether or not the height of sons is related to father’s height (x1) and
mother’s height (x2), data were gathered and part of the multiple regression excel
output is shown below. Fill the table and answer the following questions.
SUMMARY
OUTPUT
Regression Statistics
Multiple R 0.5169
R Square 0.2672
Adjusted R Square 0.2635
Standard Error 8.0683
Observations 400
ANOVA
df SS MS F Significance F
Regression 2 9421.58
4712.2
9
72.389
0 0.0000
Residual 397 25843.41
65.096
8
Total 399 35264.98
Coefficients
Standard
Error t Stat P-value
Intercept 93.8993 8.0072
11.726
9 0.0000

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X1 0.4849 0.0412
11.777
2 0.0000
X2 -0.0229 0.0395 -0.5811 0.5615
a. What is the standard error of estimate? What does this statistic tell you?
The standard error of estimate is usually referred to the measure of the accuracy of the
predictions made in the regression analysis (Draper and Smith, 2017). Thus, it is also a
measure of the dispersion or variability in the predicted scores in the regression analysis. By
considering the multiple regression results from the above excel output (From the regression
statistics table), the standard error of the estimate is 8.0683. This value tells us that the
average height of sons from the fitted regression line is about 8.06%. Also, in reference to
this value, the value suggests a better measure of the accuracy of the predictions (Smaller
variation).
b. What is the coefficient of determination? What does this statistic tell you?
The coefficient of determination in multiple regression measures the proportion of the
variation in the dependent variable that can be predicted from the set of the independent
variables in the regression equation (Woodside, 2013). For this case, the independent variable
is the son’s height while the dependent variables are mothers(X1) and father’s heights (X2)
respectively. From the above output, the coefficient of determination (R Square) is 0.2672.
This is not closer to 1 and thus the regression equation does not fit the data well. The data is
therefore not distributed well.
c. What is the adjusted coefficient of determination for the degree of freedom? What do
this statistic and the one refer to in part (b) tell you about how well the model fits the data.

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From the above output results (From the regression statistics table), the adjusted coefficient
of the determination for the degrees of freedom (Adjusted R Square) is 0.2635. Since both the
coefficient of determination and the adjusted coefficient of the determination of the degrees
of freedom are not close to one, the model thus does not fits the data well. Normally, a value
closer to 1 indicates a better fit of the data than a value not closer to 1 (Cohen et al., 2016).
d. Test the overall utility of the model. What does the test result tell you?
In testing the overall utility of the model, the F-test is applied since the F-test of the overall
significance indicates whether the linear regression model provides a better fit to the data
than a model that contains no independent variables (Todeschini et al., 2018). Thus to test the
overall utility of the model, we conduct hypothesis testing and examine the results as follows;
Stating the null and alternative hypotheses as;
Null Hypothesis (Ho): This model with independent variables fits the data well i.e.
β1 = 0 and β3 = 0
Alternative Hypothesis (Ha): This model with independent does not variables (fits
the data better than the intercept) i.e. β1 ≠ 0 and β3 ≠ 0
From the Analysis of Variance table (ANOVA table), the F-test statistics is 72.3890 with a P-
Value of 0.000. Since this P-value is less than 0.05, we reject the null hypothesis that the
regression parameters are zero at significance level 0.05. Thus it can be concluded that the
parameters are jointly statistically significant at a significance level of 0.05 and hence the
model does not fit the data well
e. Interpret each of the coefficients.
To interpret the coefficients, we consider the regression coefficient table;
The y-intercept of this regression model is 93.8993. This is the constant of the regression
model.

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The coefficient of the father’s height (x1) from the table is 0.4849. This is a positive value
coefficient. It implies that the father’s height is 0.4849 times higher than that of the son. This
also means that for every unit increase in son’s height, there is an increase in the father’s
height.
For the case of the mother’s height relation (x2), the coefficient is -0.0229. As seen, this
negative value. This suggests a relation in the opposite direction. It simply implies that when
the height of the mother increases, the height of the son decreases. The dependent variable
decreases by this coefficient.
f. Do these data allow the statistic practitioner to infer that the heights of the
sons and the fathers are linearly related?
Based on the analysis conducted and by examining the coefficient of the son’s height and the
father’s height, the value of the coefficient is obtained to be 0.4849. Since this value is a
positive one, it suggests a positive relation or association between the heights of the sons and
the father’s height. Thus these data can allow any statistics practitioner to infer that the
heights of the sons and fathers are linearly related.
Also, from the analysis of variance results (ANOVA table) with regard to the test of
hypothesis, the F=72. 8390>0.000. This shows that we have statistically significant evidence
at α = 0.05 to show that there is a difference in mean height between the sons and fathers
height. With this result, the statistical practitioners are right to infer/assert that heights of the
sons and fathers are linearly related.
g. Do these data allow the statistic practitioner to infer that the heights of the
sons and the mothers are linearly related?
In regard to this, we consider the coefficient of the mother’s height in relation to the son’s
height. From the findings above, the coefficient is obtained as -0.0229 (X2 coefficient). This

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is a negative coefficient. This suggests that there is a negative relationship between the
variables. This basically means that the height of the son is negatively related to the height of
the mother. Hence it would be appropriate for the statistics practitioner to infer that the
heights of the sons and the height of the mothers are linearly related or associated.
References
Draper, N. R., & Smith, H. (2017). Applied regression analysis (Vol. 326). John Wiley &
Sons.
Cohen, P., West, S. G., & Aiken, L. S. (2016). Applied multiple regression/correlation
analysis for the behavioral sciences. Psychology Press.
Todeschini, R., Consonni, V., Mauri, A., & Pavan, M. (2018). Detecting “bad” regression
models: multicriteria fitness functions in regression analysis. Analytica Chimica
Acta, 515(1), 199-208.
Woodside, A. G. (2013). Moving beyond multiple regression analysis to algorithms: Calling
for adoption of a paradigm shift from symmetric to asymmetric thinking in data
analysis and crafting theory. John Wiley & Sons.