Research Paper on Height & Age: Survey Shows Relationship Between Two Variables

Introduction

A survey was conducted to determine whether there is a relationship between the height and the age of students. Thirty-two students were asked about their ages and height in feet and the data recorded. The height was taken as the dependent variable where a linear regression was used to analyze the data, with age as the dependent variable. The analysis seeks to determine if there is a relationship between the two variables and if so, what the nature of the association is. The average height of the students in the survey was 5.4 feet while the average age was 22 years. The tallest student was 6 feet tall while the shortest was 4.8 feet.

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On the other hand, the oldest student in the survey was 29 years whereas the youngest student had 18 years. From this, it can be deduced that the range in age was 11 years. The range in height was 1.2 feet. The modal height was 5.7 feet while the modal age was 20 with the median age being 22 years and the median height being 5.5 feet. Another element of the data that can be considered is the standard deviation. The data of the height of the students had a standard deviation of 0.3823. The age data of the students had a standard deviation of 3.1392. The data collected is as represented in the table below.

Analysis of Data

The analysis of the data was done using the Excel software. A linear regression test was conducted between the dependent variable (height) and the independent variable (age). Linear regression has several ways of computation. However, this report focuses on only two ways, using a scatter diagram and the linear regression output. A scatter diagram was therefore first fitted as shown below.

The diagram above shows a scatter plot of height against the age of students. The scatter plot can be used to deduce several elements on linear regression. First and foremost, it gives us the linear regression equation as shown by the linear regression line. The equation also provides information about the relationship and the form of the relationship between the dependent and the independent variables. An additional feature is the R squared. This gives the percentage of the dependent variable that is explained or accounted for by the independent variable. The results above show that there is a positive relationship between the height and the age of the students. This implies that the height of the students is dependent on their ages in that, as the age of the students' increases by a single unit, their height increments by 0.0833. Hence there is a positive association between the height and the age of the students.

Furthermore, the R squared shows that 46% of the variation in height is accounted for by the students' age. On further analysis, this can give us information on the goodness of fit of the regression analysis. The intercept (3.5193) is the value of the height when the age is equivalent to or close to zero.

After plotting the scatter plot and analyzing it, it is recommended that regression analysis is done in support of the results contained in the scatter plot above. As a result, the following regression output was generated.

The above output gives various elements of regression analysis. Firstly, it provides the regression statistics which contains the number of observations which is the number of students in the survey, the multiple and the adjusted R-squared and the R squared. However, in this survey, we are only interested in the R squared since there is only one independent variable (Montgomery, Douglas, Elizabeth, and Geoffrey, 391). An ANOVA analysis is also done, and the results are as shown. This can be used mainly in hypothesis testing, for instance, to determine the significance of age as a predictor and the significance of the regression. The final segment or section gives the values of the elements of the regression equation, which is the values of the intercept, the coefficient of age. It also provides the p-values and the values of the confidence intervals of both the intercept and the dependent variable.

The above output can be used for various statistical analysis including testing varied types of hypotheses using either the probability value or the student t statistic. From this summary output, it is evident that there is a positive association between age and the dependent variable. This is because the value of age is positive. Upon fitting the linear regression model, it is found that it is consistent with the fitted equation from the scatter plot. Moreover, 46.76% of the variation in the dependent variable is explained by the age of the students, which is the same result obtained from the scatter plot analysis. To sum it all up, from the examinations conducted above on the students' data on height and age, it can confidently be concluded that the variable age is a significant predictor of the height of the students in the survey. In simple terms, this means that the height of the students depends on their ages such that a student's height can be determined given the information on his/her age. Besides, the positive relationship means that increasing the age of a student by a single unit causes an increment in the height of the student.

Applications

Linear regression analysis finds use in many fields including business, engineering, data science, and agriculture.

Predictive analytics like forecasting future chances and dangers is the most noticeable utilization of regression analysis. Demand analysis, for example, predicts the number of items which a shopper will likely buy (Harrell & Frank, 495). Be that as it may, demand isn't the primary dependent variable with regards to business. Regression analysis can go a long ways past forecasting impact on direct income. Insurance agencies vigorously depend on regression analysis to appraise the credit standing of policyholders and a conceivable number of cases in a given period.

Regression models can likewise be utilized to upgrade various processes. A factory administrator, for instance, can make a statistical model to comprehend the effect of oven temperature on the shelf life of the cookies heated in those stoves (Fox & John, 985). In a call focus, we can investigate the connection between hold up times of guests and the number of grievances. Information-driven decision making wipes out the guesswork and corporate politics from basic decision making. This improves the execution of aspects by featuring the territories that have the most extreme effect on the operational effectiveness and incomes.

Organizations today are over-burden with information on accounts, activities and client purchases. Progressively, officials are presently inclining toward information examination to settle on educated and informed choices subsequently dispensing with the intuition, instinct and gut feel (Seber, George, and Alan, 564). Regression analysis can convey a logical point to the administration of any organization. By reducing the broad measure of crude information into significant data, regression analysis drives the best approach to more brilliant and progressively exact choices. This does not imply that regression analysis is an end to supervisors' imaginative reasoning. This method goes about as an ideal tool to test a hypothesis before plunging into execution.

Historical Background

The earliest type of regression was the technique for least squares, which was published by Legendre in 1805 and by Gauss in 1809. Legendre and Gauss both applied the strategy to the issue of deciding, from cosmic perceptions, the circles of bodies about the Sun. Gauss published a further advancement of the hypothesis of least squares in 1821 including a variant of the Gauss- Markov hypothesis.

The expression "regression" was instituted by Francis Galton in the nineteenth century to portray a natural biological phenomenon. The phenomenon was that the statures of descendants of tall ancestors would in general regress down towards a typical normal. For Galton, regression had just this organic meaning. However, his work was later stretched out by Udny Yule and Karl Pearson to a progressively broad statistical context. In work by Yule and Pearson, the joint distribution of the response and explanatory factors are thought to be Gaussian. This supposition that was debilitated by R.A. Fisher in his works of 1922 and 1925. Fisher had the assumption that the conditional distribution of the response variable is Gaussian; however, the joint distribution need not be. In this regard, Fisher's supposition is nearer to Gauss' formulation of 1821.

Regression strategies keep on being a region of dynamic research. In recent decades, new techniques have been produced for robust regression, regression involving related responses, for example, growth curves and time series, non-parametric regression, Bayesian strategies for regression, regression in which the predictor factors are estimated with error, causal inference with regression, and regression with more predictor factors than observations.

Works Cited

Seber, George AF, and Alan J. Lee. Linear regression analysis. Vol. 329. John Wiley & Sons, 2012.

Montgomery, Douglas C., Elizabeth A. Peck, and G. Geoffrey Vining. Introduction to linear regression analysis. Vol. 821. John Wiley & Sons, 2012.

Harrell Jr, Frank E. Regression modeling strategies: with applications to linear models, logistic and ordinal regression, and survival analysis. Springer, 2015.

Fox, John. Applied regression analysis and generalized linear models. Sage Publications, 2015.