Relationship Between Children Age and Show Size

Question: Describe about the Relationship Between Children Age and Show Size? Answer: Introduction Here, we have to study the relationship between the age of the children and their shoes size. For checking this, we have to analyse some data for the shoes size of the children. We have to see some descriptive statistics for the shoes size for the children with different ages. Also, we have to see some graphical analysis for this study. Let us see this statistical data analysis in detail given as below: Aim To check the relationship between the age of the children and their shoe size To find the average values for the age and shoe size for the children Hypothesis and Null hypothesis Null hypothesis: There is a correlation between the shoe size and age of the children. Alternative hypothesis: There is no any correlation exists between the shoe size and the age of the children. Method Here, we have to use some statistical techniques of correlation analysis. Also, we have to see some descriptive statistics for the variable shoe size of children with different ages. Design A random sample of 20 participants is taken for the study of correlation analysis of the shoe size and the age of the children. The design of correlation and regression model is used for the study of shoe size and age of children. Participants A sample of 20 children with their ages and shoe sizes were chosen for this study. Materials Ages of the student are measured by using their birth certificates and the shoe sizes were measured by using scale. Data analysis and Results Data is collected for the 20 childrens. The method of random sampling is used while collecting the data for the childrens of different ages. Data is given as below: Participant Age Shoe size 1 5 5 2 6 5 3 8 5 4 9 6 5 7 6 6 6 5 7 8 6 8 5 5 9 4 5 10 9 6 11 8 6 12 5 5 13 6 5 14 9 6 15 10 6 16 12 8 17 12 7 18 13 8 19 14 8 20 10 6 The descriptive statistics for the variable age is given as below: Age Mean 8.3 Standard Error 0.645225379 Median 8 Mode 5 Standard Deviation 2.885535616 Sample Variance 8.326315789 Kurtosis -0.70464336 Skewness 0.426148643 Range 10 Minimum 4 Maximum 14 Sum 166 Count 20 The average age of the children is given as 8.3 years with the standard deviation of 2.89 years. The descriptive statistics for the variable show size is given as below: Shoe size Mean 5.95 Standard Error 0.234801148 Median 6 Mode 5 Standard Deviation 1.050062655 Sample Variance 1.102631579 Kurtosis 0.068287532 Skewness 1.017479158 Range 3 Minimum 5 Maximum 8 Sum 119 Count 20 The average show size for children is given as 5.95 inches with the standard deviation of 1.05 inches. The scatter diagram for the shoe size of the children and the age of the children is given as below: Ages are given in years. (Rounded) Shoe sizes are given in inches. (Rounded) The correlation coefficient is given as below: Age Shoe size Age 1 Shoe size 0.925832 1 The correlation coefficient between the age and shoe size for the student is given as 0.9258, this means, there is a high positive linear relationship or strong positive association or correlation exists between the given two variables such as age and shoe size. So we concluded that as the age of the student increases, the shoe size of the student is also increases. The regression analysis is given as below: SUMMARY OUTPUT Regression Statistics Multiple R 0.925831506 R Square 0.857163978 Adjusted R Square 0.849228643 Standard Error 0.407731847 Observations 20 ANOVA Df SS MS F Significance F Regression 1 17.95758534 17.95759 108.0186 4.91733E-09 Residual 18 2.992414665 0.166245 Total 19 20.95 Coefficients Standard Error t Stat P-value Lower 95% Intercept 3.153603034 0.284087422 11.10082 1.75E-09 2.55675751 Age 0.336915297 0.032416894 10.3932 4.92E-09 0.268809931 The value of the R square or coefficient of determination is given as 0.8572 or 85.72%, this means, about 85.72% of the variation in the dependent variable show size of the student is explained by the independent variable age of the student. Here, we get the p-value for this regression model as approximately equal to zero, therefore we reject the null hypothesis that the given regression model is significant. The regression equation for this regression model is given as below: Y = 3.1536 + 0.3369*X Where y is show size of the student and x is the age of the student. Conclusions The average age of the children is given as 8.3 years with the standard deviation of 2.89 years. The average show size for children is given as 5.95 inches with the standard deviation of 1.05 inches. The correlation coefficient is found as 0.9258 which means there is a strong association or linear relationship exists between the two variables shoe size and age of children. This means, we conclude that as the age of the student increases, the shoe size of the student also increases. The value of the R square or coefficient of determination is given as 0.8572 or 85.72%, this means, about 85.72% of the variation in the dependent variable show size of the student is explained by the independent variable age of the student. References: Leonard J. Savage, The Foundations of Statistics, 2nd ed., Dover Publications, Inc. New York, 1972. Robert V. Hogg, Allen T. Craig, Joseph W. McKean, An Introduction to Mathematical Statistics, 6th ed., Prentice Hall, 2004. George Casella, Roger L. Berger, Statistical Inference, 2nd ed., Duxbury Press, 2001. David R. Cox, D. V. Hinkley, Theoretical Statistics, Chapman Hall/CRC, 1979. Peter J. Bickel, Kjell A. Doksum, Mathematical Statistics, Volume 1, Basic Ideas and Selected Topics, 2rd ed. Prentice Hall, 2001. S. Ferguson, Mathematical Statistics: A Decision Theoretic Approach, Academic Press, Inc., New York, 1967 Harald Cramr, Mathematical Methods of Statistics, Princeton, 1946 Schervish, Mark J. (1995). Theory of statistics (Corr. 2nd print. ed.). New York: Springer Moses, Lincoln E. (1986) Think and Explain with Statistics, Addison-Wesley Hays, William Lee, (1973) Statistics for the Social Sciences, Holt, Rinehart and Winston