Question

question 3 0/1 pt 3 99 detail
based on the data shown below, calculate the regression line (each value to at least two decimal places)
y =
\bx +

x	y
2	9.7
3	13.6
4	15.4
5	15.2
6	19.1
7	22
8	21.4

question help: message instructor
submit question

Explanation:

Step1: Calculate necessary sums

First, we need to calculate \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \) for the given data.

The data points are:
\( (x_1, y_1) = (1, 8.3) \)
\( (x_2, y_2) = (2, 9.7) \)
\( (x_3, y_3) = (3, 13.6) \)
\( (x_4, y_4) = (4, 15.4) \)
\( (x_5, y_5) = (5, 15.2) \)
\( (x_6, y_6) = (6, 19.1) \)
\( (x_7, y_7) = (7, 22) \)
\( (x_8, y_8) = (8, 21.4) \)

Calculating \( \sum x \):
\( \sum x = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = \frac{8(8 + 1)}{2} = 36 \)

Calculating \( \sum y \):
\( \sum y = 8.3 + 9.7 + 13.6 + 15.4 + 15.2 + 19.1 + 22 + 21.4 \)
\( = 8.3 + 9.7 = 18; 18 + 13.6 = 31.6; 31.6 + 15.4 = 47; 47 + 15.2 = 62.2; 62.2 + 19.1 = 81.3; 81.3 + 22 = 103.3; 103.3 + 21.4 = 124.7 \)

Calculating \( \sum xy \):
\( (1)(8.3) + (2)(9.7) + (3)(13.6) + (4)(15.4) + (5)(15.2) + (6)(19.1) + (7)(22) + (8)(21.4) \)
\( = 8.3 + 19.4 + 40.8 + 61.6 + 76 + 114.6 + 154 + 171.2 \)
\( 8.3 + 19.4 = 27.7; 27.7 + 40.8 = 68.5; 68.5 + 61.6 = 130.1; 130.1 + 76 = 206.1; 206.1 + 114.6 = 320.7; 320.7 + 154 = 474.7; 474.7 + 171.2 = 645.9 \)

Calculating \( \sum x^2 \):
\( 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 \)
\( = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = \frac{8(8 + 1)(2 \times 8 + 1)}{6} = 204 \) (using the formula for sum of squares of first \( n \) natural numbers \( \sum_{i = 1}^n i^2=\frac{n(n + 1)(2n + 1)}{6} \))

Step2: Calculate the slope \( m \)

The formula for the slope \( m \) of the regression line \( y = mx + b \) is:
\( m=\frac{n\sum xy-\sum x\sum y}{n\sum x^2 - (\sum x)^2} \)
where \( n = 8 \) (number of data points)

Substitute the values:
\( n = 8 \), \( \sum xy = 645.9 \), \( \sum x = 36 \), \( \sum y = 124.7 \), \( \sum x^2 = 204 \)

First, calculate the numerator:
\( n\sum xy-\sum x\sum y=8\times645.9 - 36\times124.7 \)
\( = 5167.2-4489.2 = 678 \)

Then, calculate the denominator:
\( n\sum x^2-(\sum x)^2=8\times204 - 36^2 \)
\( = 1632 - 1296 = 336 \)

Now, find \( m \):
\( m=\frac{678}{336}\approx2.02 \) (rounded to two decimal places)

Step3: Calculate the y-intercept \( b \)

The formula for the y-intercept \( b \) is:
\( b=\frac{\sum y - m\sum x}{n} \)

Substitute the values:
\( \sum y = 124.7 \), \( m\approx2.02 \), \( \sum x = 36 \), \( n = 8 \)

First, calculate \( m\sum x \):
\( 2.02\times36 = 72.72 \)

Then, calculate \( \sum y - m\sum x \):
\( 124.7-72.72 = 51.98 \)

Now, find \( b \):
\( b=\frac{51.98}{8}\approx6.50 \) (rounded to two decimal places)

Answer:

\( y = \boldsymbol{2.02}x + \boldsymbol{6.50} \)