The following results were obtained with respect to two variable x and y:
`sumx = 30, sum y = 42, sumxy = 199, sumx^(2) = 184, sumy^(2) = 318, n=6`.
Find the following:
(i) The regression coefficients.
(ii) Correlation coefficient between x and y.
(iii) Regression equation of y on x.
(iv) The likely value of y when x = 10.

Answer

To solve the problem step by step, we will follow the instructions given in the question.

### Given Data:
- \( \Sigma x = 30 \)
- \( \Sigma y = 42 \)
- \( \Sigma xy = 199 \)
- \( \Sigma x^2 = 184 \)
- \( \Sigma y^2 = 318 \)
- \( n = 6 \)

### (i) Finding the Regression Coefficients

1. **Calculate \( b_{xy} \)** (Regression coefficient of y on x):
\[ b_{xy} = \frac{n \Sigma xy - \Sigma x \Sigma y}{n \Sigma x^2 - (\Sigma x)^2} \]
Substituting the values:
\[ b_{xy} = \frac{6 \times 199 - 30 \times 42}{6 \times 184 - 30^2} \]
\[ = \frac{1194 - 1260}{1104 - 900} \]
\[ = \frac{-66}{204} = -\frac{11}{34} \]

2. **Calculate \( b_{yx} \)** (Regression coefficient of x on y):
\[ b_{yx} = \frac{n \Sigma xy - \Sigma x \Sigma y}{n \Sigma y^2 - (\Sigma y)^2} \]
Substituting the values:
\[ b_{yx} = \frac{6 \times 199 - 30 \times 42}{6 \times 318 - 42^2} \]
\[ = \frac{1194 - 1260}{1908 - 1764} \]
\[ = \frac{-66}{144} = -\frac{11}{24} \]

### (ii) Finding the Correlation Coefficient \( r_{xy} \)

The correlation coefficient can be calculated using the formula:
\[ r_{xy} = \sqrt{b_{xy} \times b_{yx}} \]
Substituting the values:
\[ r_{xy} = \sqrt{-\frac{11}{34} \times -\frac{11}{24}} = \sqrt{\frac{121}{816}} = \frac{11}{\sqrt{816}} \approx 0.384 \]

### (iii) Regression Equation of y on x

The regression equation of y on x is given by:
\[ y - \bar{y} = b_{xy}(x - \bar{x}) \]
First, we need to calculate \( \bar{x} \) and \( \bar{y} \):
\[ \bar{x} = \frac{\Sigma x}{n} = \frac{30}{6} = 5 \]
\[ \bar{y} = \frac{\Sigma y}{n} = \frac{42}{6} = 7 \]
Now substituting into the regression equation:
\[ y - 7 = -\frac{11}{34}(x - 5) \]
Rearranging gives:
\[ y = -\frac{11}{34}x + \frac{55}{34} + 7 \]
Converting 7 to a fraction:
\[ y = -\frac{11}{34}x + \frac{55}{34} + \frac{238}{34} \]
\[ y = -\frac{11}{34}x + \frac{293}{34} \]

### (iv) Likely Value of y when \( x = 10 \)

Substituting \( x = 10 \) into the regression equation:
\[ y = -\frac{11}{34}(10) + \frac{293}{34} \]
\[ y = -\frac{110}{34} + \frac{293}{34} \]
\[ y = \frac{293 - 110}{34} = \frac{183}{34} \approx 5.38 \]

### Final Answers:
1. Regression Coefficients: \( b_{xy} = -\frac{11}{34}, b_{yx} = -\frac{11}{24} \)
2. Correlation Coefficient: \( r_{xy} \approx 0.384 \)
3. Regression Equation: \( y = -\frac{11}{34}x + \frac{293}{34} \)
4. Likely Value of y when \( x = 10 \): \( y \approx 5.38 \)