correlation between heigh and weigh

Correlation between length and weight
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Correlation between length and weight
This paper aims at discussing the statistical ways in which the relationship between the weight and length of bears can be analyzed.
The relationship weight (W) and length (L) for almost all bears can be presented by the Weight and length relationship -W = qLb (1.1).
Null Hypothesis: There is no statistical significance between length and weight
Alternative hypothesis: There is a statistical significance relationship between length and weight
Where W and L represent the weight and length of the bears while ‘q ‘and ‘b’ represent constants. The constant ‘b’ indicates the rate of weight gain in relation to the length growth or the rate at which the weight of the bear increases with an increase in length.
The constants q and b can be estimated by use of linear functions. However, as observed in ecological setups, these functional relationships tend to be non-linear. For instance, in our case, the weight-length relationship is non-linear. This nonlinear functions (Curvilinear) relationships can be easily be transformed to linear functions by converting both sides to natural logarithms (Brase & Brase, 2009).
Ln W= ln q+ b.ln L (1.2).
The above equation is equal to the linear regression equation given below.
Y=a+b*x(1.2a)
The interpretation of this equation is that; Y is equal to ln W, which represents the y-intercept (this is the point where the line cuts the y-axis).

The regression line is equivalent to ln q, b is the gradient, and x is equivalent to ln L.
This now makes it easier for one to estimate the values of a and b by use of linear regression analysis.
a = ln q
Using the antilog of a, we can calculate q of the original length-weight relationship using the formula, q = exp a (Note: exp is the inverse of ln, the base of the natural system of logarithms and equal to 2.718282).
With this formula, the estimated relationship between W (in g) and L (in cm) which is equal to W = q.Lb could be easily determined.
Descriptive statistics.
A hypothetical data with the mean length of 58.62 cm and mean weight of 182.88 kg is used to how the relationship between length and weight of bears is analyzed using the excel spreadsheet. Other useful characteristics of the two variables and the raw data are described below.
LengthWeight
Median 60.75150
Mode 72166
SD10.70121.80
Variance 114.5014835.53
The value of b=0.00826
Estimation of the value of a.
a=y-bx where X and Y are the arithmetic means of ln L and lnW respectively.
A=4.967-(0.00826*4.053)
=4.933
The linear regression now becomes:
Y=4.933+0.00826x
The above equation is equal to the function ln W = ln q + b*ln L hence a = ln q = 4.933
To obtain q, find the antilog of a: a: q = exp a = 2.718282 (4.933) = 138.7953
To find the relationship between weight and, substitute the q in the equation W=q.Lb
W = 138.7953*L0.00826
Finding the correlation coefficient.
Correlation refers to the degree to which two different variables relate. The degree of association is expressed by the correlation coefficient which takes values ranging from +1 and -1. When the coefficient r is negative, the interpretation is that one variable (either x or y) tends to decrease while other increases are implying the existence of a negative correlation or reveres relationship between the two variables (Ross, 2004). Conversely, in situations where r is positive, it implies a direct relationship between the two variables, in that, when one variable increases, the other variable increases with the same degree. In a nutshell, the correlation coefficient is said to be higher when the value is close to +1 or -1, and smaller when the value is close to zero.
In this example, we can then calculate the correlation coefficient of the two variables (length and weight of bears). By doing this, we could have clearly achieved the objective of analyzing the degree of relationship between length and weight of the bears.
Inferential statistics
From the analysis above,
Y=4.933+0.00826x
H0: b=0.0082
H1: b~=0.0082
Using excel to analyze the data the following output is obtained.
SUMMARY OUTPUT Regression Statistics Multiple R 0.937612 R Square 0.879117 Adjusted R Square 0.876792 Standard Error 0.260995 Observations 54 ANOVA   df SS MS F Significance F Regression 1 25.76014115 25.76014115 378.1664521 1.6228E-25 Residual 52 3.542163331 0.068118526 Total 53 29.30230448         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept -9.52276 0.745974966 -12.76551574 1.18667E-17 -11.01966345 -8.02585 -11.01966345 -8.025846886
X Variable 1 3.574986 0.18383696 19.44650231 1.6228E-25 3.206090019 3.943882 3.206090019 3.943881716
The p-value based on the analysis is 1.6228. Since the p >0.05, the results of the analysis are statistically insignificant, and we fail to accept the null hypothesis. The gradient of the two variables is, therefore, greater than 0.0082, and we accept the alternative hypothesis.

The graph expresses the association between the length (x-axis) and the weight (y-axis) of the bears before conversion to the natural logarithms.

In conclusion, the graph represents the relationship between the length and weight of the bears using data values that have been converted to their natural logs. Lack of association between the given variables may result from the variability of food in the ecological setup where the bears reside. Other factors that may lead to the variability include the sex of the bears including their gonad developments and also the species of the bears.
References
Brase, C. & Brase, C. (2009). Understandable statistics (1st ed.). Boston, Mass.: BrooksCole.
Ross, S. (2004). Introduction to probability and statistics for engineers and scientists (1st ed.). Amsterdam: Elsevier/Academic Press.