Answer to Question #228248 in Statistics and Probability for Himanshu

Question

1(a) After settlement, the average weekly wage in a factory had increased from Rs. 8 to 12

and standard deviation had increased from Rs. 1 to 1.5. The wages have become higher and more

uniform, after settlement. Comment.

1(b) Calculate the Karl Pearson’s coefficient of correlation from the following pairs of values

and interpret the result:

Values of X 12 9 8 10 11 13 7

Values of Y 14 8 6 9 11 12 3

1(c) Distinguish between correlation and regression.

1(d) The two regression coefficients are -2.7 and -0.3 and the coefficient of correlation is 0.90.

Comment.

Answer 1

Part 1a

Coefficient of Variation = (Standard deviation /Mean) * 100 %

Earlier Average weekly wages = 8

Mean = 8

Standard Deviation = 1

Coefficient of Variation = $(\frac{1}{8})*100 = 12.5 \%$

After settlement, Average weekly wages = 12

Mean = 12

Standard Deviation = 1.5

Coefficient of Variation = $(\frac{1.5}{12})*100 = 12.5 \%$

12 > 8

Hence wages have become higher is Correct

Coefficient of Variation is same

hence no impact on the uniformity

wages have become more uniform is not correct

wages have become higher is Correct but more uniform is not correct

Part 1b

$$\bar{X}=\dfrac{\sum_iX_i}{n}=\dfrac{70}{7}=10$$ $$\bar{Y}=\dfrac{\sum_iY_i}{n}=\dfrac{63}{7}=9$$ $$SS_{XX}=\sum_i(X_i-\bar{X})^2=\sum_iX_i^2-n\cdot\bar{X}^2$$ $$=728-7(10)^2=28$$ $$SS_{YY}=\sum_i(Y_i-\bar{Y})^2=\sum_iY_i^2-n\cdot\bar{Y}^2$$ $$=651-7(9)^2=84$$

$$SS_{XY}=\sum_i(X_i-\bar{X})(Y_i-\bar{Y})=\sum_iX_iY_i-n\cdot\bar{X}\bar{Y}$$ $$=676-7(10)(9)=46$$ $$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}=\dfrac{46}{\sqrt{28}\sqrt{84}}\approx$$ $$\approx0.948504$$

$0.7<r\leq1$ means a strong positive correlation.

Part 1c

The key distinction between correlation and regression is that the former evaluates the strength of a connection between two variables, let us say x and y. In this context, correlation is used to quantify degree, whereas regression is a metric used to determine how one variable influences another.

Part 1d

Regression coefficients are -2.7 and -0.3

The coefficient in regression with a single independent variable shows you how much the dependent variable is anticipated to grow (if the coefficient is positive) or decrease (if the coefficient is negative) when the independent variable increases by 2.7 and -0.3.

The coefficient of correlation is 0.90

A correlation of -0.90, for example, implies the same degree of clustering as a correlation of +0.90. A positive correlation indicates that the cloud slopes upward; when one variable rises, the other rises as well. A negative correlation indicates that the cloud slopes downward; as one variable rise, the other falls.