Question

Below is a small data set showing observations on two variables X and Y. X Y 0.408 0.173 0.275 0.084 0.375 0.115 0.349 0.156 0.312 0.070 0.382 0.210 We are especially interested in determining if there is a linear relationship between Y and X (Y is the dependent variable and X is the explanatory variable). a) Begin by making a scatterplot of Y vs X. b) Estimate a linear equation for predicting Y given X. c) How good is your equation? d) Calculate the appropriate 'summary ANOVA TABLE e) Calculate R? Interpret Rand state what can be concluded from your work to this point.

Answer 1

a)

Scatter Plot Yvs X 0 0.1 0.2 0.3 0.4 0.5

A scatter plot (also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram) is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data. A relationship is linear when the points on a scatterplot follow a somewhat straight line pattern. In the above diagram we can see the data points follow a somewhat straight line pattern.

The scatterplot shows that X and Y are positively correlated.

b) Y=mX+b

m=Slope= (Y2-Y1/X2-X1) or can be calculated using slope function in excel.

b= intercept (can be calculated using intercept function in excel.)

Linear Equation is : Y= -0.16 + 0.85X

0.408 0.275 0.375 0.349 0.312 0.382 0.173 0.084 0.115 0.156 0.07 0.21 Slope Y-intercep 0.858932 -0.1661

C) The equation between two variables gives a straight line when plotted. Hence, it is good.

D) Analysis of variance or ANOVA can be used to compare the means between two or more groups of values. We can test the null hypothesis that the means of each sample are equal against the alternative that not all the sample means are the same.

E)

Anova: Single Factor SUMMARY Groups Count Sum Average Variance 6 2.101 0.350166667 0.002416567 6 0.808 0.134666667 0.002947067 ANOVA Source of Variation Between Groups Within Groups SS I 0.13932075 0.026818167 df MS P-value Fcrit 1 0.13932075 51.95013952 2.89981E-05 4.964602744 10 0.002681817 Total 0.166138917 11

R-squared is a statistical measure of how close the data are to the fitted regression line. It is also known as the coefficient of determination, or the coefficient of multiple determination for multiple regression.

R-Square can be calculated using the below formula or by using Excel (Data Analysis - Regression- Specify X and Y values).

R Squared Formula = r2 n (Σ Xy) - (Σκ)(Σy) V[nΣx - (Σκ?][nsy' - (Σy] Γ =

R-squared = Explained variation / Total variation

SUMMARY OUTPUT Regression Statistics Multiple R 0.777791586 R Square 0.604959751 Adjusted R Square 0.506199688 Standard Error 0.038147902 Observations

R-squared is always between 0 and 100%:

• 0% indicates that the model explains none of the variability of the response data around its mean.
• 100% indicates that the model explains all the variability of the response data around its mean.

In general, the higher the R-squared, the better the model fits your data.

0.60 R square value indicates that means that 60 percent of the variation in the Y data is due to variation in the X data.