Relationship Between Son’s Height, Father’s Height And Mother’s Height – Regression Analysis

Frequency distribution of marks

1. a) An online survey method could be used for data collection considering the fact that the underlying questions are straight forward and also obtaining the selected random sample through face to face or other means may be difficult. It would make sense to have a higher sample fill in the online survey in relation to the preparation hours and the marks scored and then through technology enabled tools, the requisite sample of 100 can be obtained (Hillier, 2016).
2. b) The requisite sampling method which would be used to select the sample would be stratified random sampling. This would be preferred over simple random sampling so as to ensure that key attributes such as gender, educational background, country of origin and other aspects could be taken care of and a sample that is representative of the population may be obtianed. Using a simple random sample instead could lead to the sample being non-representative as certain attributes may be over-represented while other under-represented (Flick, 2015).
3. c) The independent variable is the amount of preparation time that each student spends while the dependent variable is the number of marks scored in exam. This is because typcially the marks scored would be dependent on the amount of preparation that is done by the students and not the other way around. Both the given data are numerical in nature and the measurement scale would be ratio considering the absolute zero can be defined for both variables (Eriksson & Kovalainen, 2015).
4. d) Potential issues that may be faced with regards to collection of data are highlighted below (Medhi, 2016).

• It is possible that students may not have a fair estiamte of the exact preparation time and also the time frame over which the same has to be stated. For instance, should be include 24 hours before the exam or a week or a month before the exam.
• Also, it might be possible that students may tend to overestiamte and underestiamte their study hours. For instance, students with good marks are likely to reportn higher study hours as compared to those who have lower marks in exam.

1. e) The frequency distribution of preparation time is indicated below.

From the above histogram, it is apparent that the distribution is assymetric and also there is present of skew on the left considering the fact that tail on the left seems longer than the one on the right. As a result, it is apparent that the distribution of preparation time is not normally distributed (Fehr & Grossman, 2013).

The frequency distribution of the marks is indicated below.

From the above histogram, it is apparent that the distribution is assymetric and also there is present of skew on the left considering the fact that tail on the left seems longer than the one on the right. As a result, it is apparent that the distribution of exam marks is not normally distributed (Flick,. 2015).

1. f) The requisite scatter plot is indicated below.The independent variable (i.e. preparation time) is on X axis while the dependent variable (i.e. mark) is on Y axis.

1. g) The equation of the estimated fitting line is shown below.

Mark = 28.984 + 0.5831*Preparation Time

If the preparation time would increase by 1 hour, then the marks would increase by 0.583.

1. h) The requisite numerical summary report is indicated below.i) The linear relationship between the given two variables would be indicated by the correlation coefficient whose value has come out as 0.5466.

Based on the above, it is appropriate to conlcude that the two variables have a positive relationship owing to the positive sign of the correlation coefficient. Also, this relationship is moderately strong considering that it is greater 0.5 and the theoretical maximum is 1 (Hastie, Tibshirani & Friedman, 2014).

The completed table is shown as follows.

• From the above table, standard error of estimate is 8.0683. This is indicative of the deviation between the dependent variable (i.e. son height) predicted values based on the regression model and the actual values (Hillier, 2016).
• R²is 0.2672. This is indicative that independent variables jointly can account for 26.72% of the variation in the son’s height (dependent variable). As a result, the remaining variation is unexplained by the given multiple regression model (Flick, 2015).
• Adjusted R²value is 0.2635. Taking into consideration both R² and adjusted R², it may be concluded that the multiple regression model presents a poor fit. This may be on account of a given slope coefficient being insignificant and also the low predictive power of the regression model (Medhi, 2016).The requisite hypotheses are indicated below.The significance level has been assumed as 5%.

Considering the table above, the relevant information is summarised below.

F statistic = (4710.79/65.10) = 72.336

Based on the above, the p value has come out as 0.00.

As P value < Level of Significance, H₀ is rejected but H₁ is accepted (Hair et. al., 2015). The conclusion can be drawn that the multiple regression model is statistically significant as there exists atleast one slope coefficient which is non-zero (Flick, 2015).

• The slope coefficient can be interpreted in the following manner.

 X1 Slope coefficient: It indicates that a change in father’s height by 1 unit would bring about a corresponding change in the son’s height by 0.48 units and both the changes would be in the same direction.

X2 Slope coefficient: It indicates that a change in mother’s height by 1 unit would bring about a corresponding change in the son’s height by 0.02 units and both the changes would be in the opposite direction.f) The requisite hypotheses to be tested are summarised as follows.

On the basis of the provided regression output, it becomes evident that the X1 variable slope coefficient is significant owing to the fact that corresponding p value amounts to 0.000 and therefore would not exceeds the level of significance. The net result would be rejection of null hypothesis and acceptance of alternative hypothesis. Therefore, it is correct to conclude the son’s height is related to their father’s height.

1. g) The requisite hypotheses to be tested are summarised as follows.On the basis of the provided regression output, it becomes evident that the X2 variable slope coefficient is not significant owing to the fact that corresponding p value amounts to 0.562 and therefore would exceed the level of significance. The net result would be non-rejection of null hypothesis and non-acceptance of alternative hypothesis (Lieberman, Nag, Hiller & Basu, 2013). Therefore, it is correct to conclude the son’s height is not related to their mother’s height.

References

Eriksson, P. & Kovalainen, A. (2015) Quantitative methods in business research. 3rd ed. London: Sage Publications.

Fehr, F. H. & Grossman, G. (2013). An introduction to sets, probability and hypothesis testing. 3rd ed. Ohio: Heath.

Flick, U. (2015) Introducing research methodology: A beginner’s guide to doing a research project. 4th ed. New York: Sage Publications.

Hair, J. F., Wolfinbarger, M., Money, A. H., Samouel, P., & Page, M. J. (2015) Essentials of business research methods. 2nd ed. New York: Routledge.

Hastie, T., Tibshirani, R. & Friedman, J. (2014) The Elements of Statistical Learning. 4th ed. New York: Springer Publications.

Hillier, F. (2016) Introduction to Operations Research. 6th ed. New York: McGraw Hill Publications.

Lieberman, F. J., Nag, B., Hiller, F.S. & Basu, P. (2013) Introduction To Operations Research. 5th ed. New Delhi: Tata McGraw Hill Publishers.

Medhi, J. (2016) Statistical Methods: An Introductory Text. 4th ed. Sydney: New Age International.