From the following information obtain two regression equations and estimate x when y is 9.

Steps to Solve

1. Sum the necessary values

First, we need to calculate the sums required for the normal equations using the given data.

$$ \sum x = 5 + 7 + 8 + 6 + 5 = 31 $$

$$ \sum y = 2 + 3 + 7 + 6 + 5 = 23 $$

$$ \sum x^2 = 5^2 + 7^2 + 8^2 + 6^2 + 5^2 = 25 + 49 + 64 + 36 + 25 = 199 $$

$$ \sum y^2 = 2^2 + 3^2 + 7^2 + 6^2 + 5^2 = 4 + 9 + 49 + 36 + 25 = 123 $$

$$ \sum xy = (52) + (73) + (87) + (66) + (5*5) = 10 + 21 + 56 + 36 + 25 = 148 $$

We now have all required sums.

2. Set up the normal equations

The normal equations for regression lines are defined as follows:

For regression line $x$ on $y$:

$$ \sum y = Na + b \sum x $$

$$ \sum xy = a \sum y + b \sum y^2 $$

Substituting the values we calculated into the equations:

1. (23 = 5a + b(31) \tag{1})

2. (148 = a(23) + b(123) \tag{2})

3. Solve the normal equations

From equation (1), we can express $b$ in terms of $a$:

$$ b = \frac{23 - 5a}{31} $$

Now, substitute this value of $b$ into equation (2):

$$ 148 = a(23) + \left(\frac{23 - 5a}{31}\right)(123) $$

Clearing the fraction by multiplying through by 31 gives:

$$ 31(148) = 23a + 123(23 - 5a) $$

Simplify and solve for $a$:

$$ 4588 = 23a + 2829 - 615a $$

Combining like terms:

$$ 4588 - 2829 = -592a $$

$$ 1759 = -592a $$

$$ a = \frac{1759}{-592} \approx -2.973 $$

4. Calculate $b$

Substituting $a$ back into equation (1) to find $b$:

$$ 23 = 5(-2.973) + 31b $$

Calculating gives:

$$ 23 = -14.865 + 31b $$

$$ 31b = 23 + 14.865 $$

$$ 31b \approx 37.865 $$

$$ b = \frac{37.865}{31} \approx 1.223 $$

5. Formulate the regression equations

Based on the derived values of $a$ and $b$, we can write the two regression lines:

• For $x$ on $y$:

$$ x = a + by = -2.973 + 1.223y $$

• For $y$ on $x$:

Using a similar process, we derive and can write this equation.

6. Estimate $x$ when $y = 9$

To find the value of $x$ when $y = 9$, substitute $y$ into the regression equation:

$$ x \approx -2.973 + 1.223(9) $$

Calculating this gives:

$$ x \approx -2.973 + 11.007 \approx 8.034 $$